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INTRODUCTION TO TRIGONOMETRY

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CLASS X : CHAPTER – 8 INTRODUCTION TO TRIGONOMETRYIMPORTANT FORMULAS & CONCEPTSMCQ WORKSHEET-IMCQ WORKSHEET-IIMCQ WORKSHEET-IIIMCQ WORKSHEET-IVPRACTICE QUESTIONS: TRIGONOMETRIC RATIOSPRACTICE QUESTIONS: T – RATIOS OF PARTICULAR ANGLESPRACTICE QUESTIONS: COMPLEMENTARY ANGLESPRACTICE QUESTIONS: TRIGONOMETRIC IDENTITIES

CLASS X : CHAPTER – 8 INTRODUCTION TO TRIGONOMETRY

IMPORTANT FORMULAS & CONCEPTS

Trigonometry: Derived from Greek words ‘tri’ (three), ‘gon’ (sides), and ‘metron’ (measure), it is the study of relationships between the sides and angles of a triangle.

Trigonometric Ratios (T-Ratios) of an acute angle:

In a right-angled triangle, if we consider an acute angle \(\theta\):

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  • \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)
  • \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)
  • \( \tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \)
  • \( \text{cosec } \theta = \frac{\text{Hypotenuse}}{\text{Opposite Side}} \)
  • \( \sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} \)
  • \( \cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} \)

Reciprocal Relations:

  • \( \sin \theta = \frac{1}{\text{cosec } \theta} \), \( \text{cosec } \theta = \frac{1}{\sin \theta} \)
  • \( \cos \theta = \frac{1}{\sec \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \)
  • \( \tan \theta = \frac{1}{\cot \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \)

Quotient Relations:

  • \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
  • \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Trigonometric Ratios of Complementary Angles:

  • \( \sin (90^\circ – \theta) = \cos \theta \)
  • \( \cos (90^\circ – \theta) = \sin \theta \)
  • \( \tan (90^\circ – \theta) = \cot \theta \)
  • \( \cot (90^\circ – \theta) = \tan \theta \)
  • \( \sec (90^\circ – \theta) = \text{cosec } \theta \)
  • \( \text{cosec } (90^\circ – \theta) = \sec \theta \)

Trigonometric Identities:
An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle involved.

  1. \( \sin^2 \theta + \cos^2 \theta = 1 \)
  2. \( 1 + \tan^2 \theta = \sec^2 \theta \)
  3. \( 1 + \cot^2 \theta = \text{cosec}^2 \theta \)

Trigonometric Ratios Table:

\(\angle A\) \(0^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(90^\circ\)
\(\sin A\) 0 \( \frac{1}{2} \) \( \frac{1}{\sqrt{2}} \) \( \frac{\sqrt{3}}{2} \) 1
\(\cos A\) 1 \( \frac{\sqrt{3}}{2} \) \( \frac{1}{\sqrt{2}} \) \( \frac{1}{2} \) 0
\(\tan A\) 0 \( \frac{1}{\sqrt{3}} \) 1 \( \sqrt{3} \) Not defined
\(\text{cosec } A\) Not defined 2 \( \sqrt{2} \) \( \frac{2}{\sqrt{3}} \) 1
\(\sec A\) 1 \( \frac{2}{\sqrt{3}} \) \( \sqrt{2} \) 2 Not defined
\(\cot A\) Not defined \( \sqrt{3} \) 1 \( \frac{1}{\sqrt{3}} \) 0

MCQ WORKSHEET-I

  1. In \(\Delta OPQ\), right-angled at P, \(OP = 7\) cm and \(OQ – PQ = 1\) cm, then the values of \(\sin Q\).
    1. 7/25
    2. 24/25
    3. 1
    4. none of these
  2. If \(\sin A = \frac{24}{25}\), then the value of \(\cos A\) is
    1. 7/25
    2. 24/25
    3. 1
    4. none of these
  3. In \(\Delta ABC\), right-angled at B, \(AB = 5\) cm and \(\angle ACB = 30^\circ\) then the length of the side BC is
    1. \(5\sqrt{3}\)
    2. \(2\sqrt{3}\)
    3. 10 cm
    4. none of these
  4. In \(\Delta ABC\), right-angled at B, \(AB = 5\) cm and \(\angle ACB = 30^\circ\) then the length of the side AC is
    1. \(5\sqrt{3}\)
    2. \(2\sqrt{3}\)
    3. 10 cm
    4. none of these
  5. The value of \(\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ}\) is
    1. \(\sin 60^\circ\)
    2. \(\cos 60^\circ\)
    3. \(\tan 60^\circ\)
    4. \(\sin 30^\circ\)
  6. The value of \(\frac{1 – \tan^2 45^\circ}{1 + \tan^2 45^\circ}\) is
    1. \(\tan 90^\circ\)
    2. 1
    3. \(\sin 45^\circ\)
    4. 0
  7. \(\sin 2A = 2 \sin A\) is true when \(A =\)
    1. \(0^\circ\)
    2. \(30^\circ\)
    3. \(45^\circ\)
    4. \(60^\circ\)
  8. The value of \(\frac{2 \tan 30^\circ}{1 – \tan^2 30^\circ}\) is
    1. \(\sin 60^\circ\)
    2. \(\cos 60^\circ\)
    3. \(\tan 60^\circ\)
    4. \(\sin 30^\circ\)
  9. \(9 \sec^2 A – 9 \tan^2 A =\)
    1. 1
    2. 9
    3. 8
    4. 0
  10. \((1 + \tan A + \sec A) (1 + \cot A – \text{cosec } A) =\)
    1. 0
    2. 1
    3. 2
    4. -1
  11. \((\sec A + \tan A) (1 – \sin A) =\)
    1. \(\sec A\)
    2. \(\sin A\)
    3. \(\text{cosec } A\)
    4. \(\cos A\)
  12. \(\frac{1 + \tan^2 A}{1 + \cot^2 A} =\)
    1. \(\sec^2 A\)
    2. -1
    3. \(\cot^2 A\)
    4. \(\tan^2 A\)

MCQ WORKSHEET-II

  1. If \(\sin 3A = \cos (A – 26^\circ)\), where 3A is an acute angle, find the value of A.
    1. \(29^\circ\)
    2. \(30^\circ\)
    3. \(26^\circ\)
    4. \(36^\circ\)
  2. If \(\tan 2A = \cot (A – 18^\circ)\), where 2A is an acute angle, find the value of A.
    1. \(29^\circ\)
    2. \(30^\circ\)
    3. \(26^\circ\)
    4. none of these
  3. If \(\sec 4A = \text{cosec } (A – 20^\circ)\), where 4A is an acute angle, find the value of A.
    1. \(22^\circ\)
    2. \(25^\circ\)
    3. \(26^\circ\)
    4. none of these
  4. The value of \(\tan 48^\circ \tan 23^\circ \tan 42^\circ \tan 67^\circ\) is
    1. 1
    2. 9
    3. 8
    4. 0
  5. If \(\Delta ABC\) is right angled at C, then the value of \(\cos(A + B)\) is
    1. 0
    2. 1
    3. 1/2
    4. n.d.
  6. The value of the expression \(\left[\frac{\sin^2 22^\circ + \sin^2 68^\circ}{\cos^2 22^\circ + \cos^2 68^\circ} + \sin^2 63^\circ + \cos 63^\circ \sin 27^\circ\right]\) is
    1. 3
    2. 0
    3. 1
    4. 2
  7. If \(\cos A = \frac{24}{25}\), then the value of \(\sin A\) is
    1. 7/25
    2. 24/25
    3. 1
    4. none of these
  8. If \(\Delta ABC\) is right angled at B, then the value of \(\cos(A + C)\) is
    1. 0
    2. 1
    3. 1/2
    4. n.d.
  9. If \(\tan A = \frac{4}{3}\), then the value of \(\cos A\) is
    1. 3/5
    2. 4/3
    3. 1
    4. none of these
  10. If \(\Delta ABC\) is right angled at C, in which \(AB = 29\) units, \(BC = 21\) units and \(\angle ABC = \alpha\). Determine the values of \(\cos^2 \alpha + \sin^2 \alpha\).
    1. 0
    2. 1
    3. 1/2
    4. n.d.
  11. In a right triangle ABC, right-angled at B, if \(\tan A = 1\), then the value of \(2 \sin A \cos A =\)
    1. 0
    2. 1
    3. 1/2
    4. n.d.
  12. Given \(15 \cot A = 8\), then \(\sin A =\)
    1. 3/5
    2. 4/3
    3. 1
    4. none of these

MCQ WORKSHEET-III

  1. In a triangle PQR, right-angled at Q, \(PR + QR = 25\) cm and \(PQ = 5\) cm, then the value of \(\sin P\) is
    1. 7/25
    2. 24/25
    3. 1
    4. none of these
  2. In a triangle PQR, right-angled at Q, \(PQ = 3\) cm and \(PR = 6\) cm, then \(\angle QPR =\)
    1. \(0^\circ\)
    2. \(30^\circ\)
    3. \(45^\circ\)
    4. \(60^\circ\)
  3. If \(\sin (A – B) = \frac{1}{2}\) and \(\cos(A + B) = \frac{1}{2}\), then the value of A and B, respectively are
    1. \(45^\circ\) and \(15^\circ\)
    2. \(30^\circ\) and \(15^\circ\)
    3. \(45^\circ\) and \(30^\circ\)
    4. none of these
  4. If \(\sin (A – B) = 1\) and \(\cos(A + B) = 1\), then the value of A and B, respectively are
    1. \(45^\circ\) and \(15^\circ\)
    2. \(30^\circ\) and \(15^\circ\)
    3. \(45^\circ\) and \(30^\circ\)
    4. none of these
  5. If \(\tan (A – B) = \frac{1}{\sqrt{3}}\) and \(\tan (A + B) = \sqrt{3}\), then the value of A and B, respectively are
    1. \(45^\circ\) and \(15^\circ\)
    2. \(30^\circ\) and \(15^\circ\)
    3. \(45^\circ\) and \(30^\circ\)
    4. none of these
  6. If \(\cos (A – B) = \frac{\sqrt{3}}{2}\) and \(\sin (A + B) = 1\), then the value of A and B, respectively are
    1. \(45^\circ\) and \(15^\circ\)
    2. \(30^\circ\) and \(15^\circ\)
    3. \(60^\circ\) and \(30^\circ\)
    4. none of these
  7. The value of \(2\cos^2 60^\circ + 3\sin^2 45^\circ – 3\sin^2 30^\circ + 2\cos^2 90^\circ\) is
    1. 1
    2. 5
    3. 5/4
    4. none of these
  8. \(\sin 2A = 2 \sin A \cos A\) is true when \(A =\)
    1. \(0^\circ\)
    2. \(30^\circ\)
    3. \(45^\circ\)
    4. any angle
  9. \(\sin A = \cos A\) is true when \(A =\)
    1. \(0^\circ\)
    2. \(30^\circ\)
    3. \(45^\circ\)
    4. any angle
  10. If \(\sin A = \frac{1}{2}\), then the value of \(3\cos A – 4\cos^3 A\) is
    1. 0
    2. 1
    3. 1/2
    4. n.d.
  11. If \(3\cot A = 4\), then the value of \(\cos^2 A – \sin^2 A\) is
    1. 3/4
    2. 7/25
    3. 1/2
    4. 24/25
  12. If \(3\tan A = 4\), then the value of \(\frac{3\sin A + 2\cos A}{3\sin A – 2\cos A}\) is
    1. 1
    2. 7/25
    3. 3
    4. 24/25

MCQ WORKSHEET-IV

  1. Value of \(\theta\), for \(\sin 2\theta = 1\), where \(0 < \theta < 90^\circ\) is:
    1. \(30^\circ\)
    2. \(60^\circ\)
    3. \(45^\circ\)
    4. \(135^\circ\)
  2. Value of \(\sec^2 26^\circ – \cot^2 64^\circ\) is:
    1. 1
    2. -1
    3. 0
    4. 2
  3. Product \(\tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \dots \tan 89^\circ\) is:
    1. 1
    2. -1
    3. 0
    4. 90
  4. \(\sqrt{1 + \tan^2 \theta}\) is equal to:
    1. \(\cot \theta\)
    2. \(\cos \theta\)
    3. \(\text{cosec } \theta\)
    4. \(\sec \theta\)
  5. If \(A + B = 90^\circ\), \(\cot B = \frac{3}{4}\) then \(\tan A\) is equal to:
    1. 3/4
    2. 4/3
    3. 1/4
    4. 1/3
  6. Maximum value of \(\frac{1}{\text{cosec } \theta}\), \(0 < \theta < 90^\circ\) is:
    1. 1
    2. -1
    3. 2
    4. 1/2
  7. If \(\cos 3\theta = \frac{\sqrt{3}}{2}\), \(\sin 2\phi = \frac{1}{2}\) then value of \(\theta + \phi\) is
    1. \(30^\circ\)
    2. \(60^\circ\)
    3. \(90^\circ\)
    4. \(120^\circ\)
  8. If \(\sin (A + B) = 1 = \cos(A – B)\) then
    1. \(A = B = 90^\circ\)
    2. \(A = B = 0^\circ\)
    3. \(A = B = 45^\circ\)
    4. \(A = 2B\)
  9. The value of \(\sin 60^\circ \cos 30^\circ – \cos 60^\circ \sin 30^\circ\) is
    1. 1
    2. -1
    3. 0
    4. none of these
  10. The value of \(2\sin^2 30^\circ – 3\cos^2 45^\circ + \tan^2 60^\circ + 3\sin^2 90^\circ\) is
    1. 1
    2. 5
    3. 0
    4. none of these

PRACTICE QUESTIONS: TRIGONOMETRIC RATIOS

  1. If \(\tan \theta = \frac{1}{\sqrt{5}}\), what is the value of \(\frac{\text{cosec}^2 \theta – \sec^2 \theta}{\text{cosec}^2 \theta + \sec^2 \theta}\)?
  2. If \(\sin \theta = \frac{4}{5}\), find the value of \(\frac{\sin \theta \tan \theta – 1}{2 \tan^2 \theta}\).
  3. If \(\cos A = \frac{1}{2}\), find the value of \(\frac{2 \sec A}{1 + \tan^2 A}\).
  4. If \(\sin \theta = \frac{\sqrt{3}}{2}\), find the value of all T–ratios of \(\theta\).
  5. If \(\cos \theta = \frac{7}{25}\), find the value of all T–ratios of \(\theta\).
  6. If \(\tan \theta = \frac{15}{8}\), find the value of all T–ratios of \(\theta\).
  7. If \(\cot \theta = 2\), find the value of all T–ratios of \(\theta\).
  8. If \(\text{cosec } \theta = \sqrt{10}\), find the value of all T–ratios of \(\theta\).
  9. If \(\tan \theta = \frac{4}{3}\), show that \(\sin \theta + \cos \theta = \frac{7}{5}\).
  10. If \(\sec \theta = \frac{5}{4}\), show that \(\frac{\sin \theta – 2\cos \theta}{\tan \theta – \cot \theta} = \frac{12}{7}\).
  11. If \(\tan \theta = \frac{1}{\sqrt{7}}\), show that \(\frac{\text{cosec}^2 \theta – \sec^2 \theta}{\text{cosec}^2 \theta + \sec^2 \theta} = \frac{3}{4}\).
  12. If \(\cos \theta = \frac{1}{2}\), show that \(\frac{1}{2}\left(\frac{\sin \theta}{\cot \theta} + \frac{2}{1 + \cos \theta}\right) = \sqrt{2}\). (Note: This question might have a typo in source, interpreted best effort).
  13. If \(\sec \theta = \frac{5}{4}\), verify that \(\frac{\tan \theta}{1 + \tan^2 \theta} = \frac{\sin \theta}{\sec \theta}\).
  14. If \(\cos \theta = 0.6\), show that \((5\sin \theta – 3\tan \theta) = 0\).
  15. In a triangle ACB, right-angled at C, in which AB = 29 units, BC = 21 units and \(\angle ABC = \theta\). Determine the values of (i) \(\cos^2 \theta + \sin^2 \theta\) (ii) \(\cos^2 \theta – \sin^2 \theta\).
  16. In a triangle ABC, right-angled at B, in which AB = 12 cm and BC = 5 cm. Find the value of \(\cos A\), \(\text{cosec } A\), \(\cos C\) and \(\text{cosec } C\).
  17. In a triangle ABC, \(\angle B = 90^\circ\), AB = 24 cm and BC = 7 cm. Find (i) \(\sin A, \cos A\) (ii) \(\sin C, \cos C\).

PRACTICE QUESTIONS: T – RATIOS OF PARTICULAR ANGLES

Evaluate each of the following:

  1. \(\sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ\)
  2. \(\cos 60^\circ \cos 30^\circ – \sin 60^\circ \sin 30^\circ\)
  3. \(\cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ\)
  4. \(\sin 60^\circ \sin 45^\circ – \cos 60^\circ \cos 45^\circ\)
  5. \(\frac{\sin 30^\circ}{\cos 45^\circ} + \frac{\cot 45^\circ}{\sec 60^\circ} – \frac{\sin 60^\circ}{\tan 45^\circ} – \frac{\cos 30^\circ}{\sin 90^\circ}\)
  6. \(\frac{\tan^2 60^\circ + 4\cos^2 45^\circ + 3\sec^2 30^\circ + 5\cos^2 90^\circ}{\text{cosec } 30^\circ + \sec 60^\circ – \cot^2 30^\circ}\)
  7. \(4(\sin^4 30^\circ + \cos^4 60^\circ) – 3(\cos^2 45^\circ – \sin^2 90^\circ) + 5\cos^2 90^\circ\)
  8. \(\frac{4}{\cot^2 30^\circ} + \frac{1}{\sin^2 30^\circ} – 2\cos^2 45^\circ – \sin^2 0^\circ\)
  9. \(\frac{1}{\cos^2 30^\circ} + \frac{1}{\sin^2 30^\circ} – \frac{1}{2} \tan^2 45^\circ – 8\sin^2 90^\circ\)
  10. \(\cot^2 30^\circ – 2\cos^2 30^\circ – \frac{3}{4}\sec^2 45^\circ + \frac{1}{4}\text{cosec}^2 30^\circ\)
  11. \((\sin^2 30^\circ + 4\cot^2 45^\circ – \sec^2 60^\circ)(\text{cosec}^2 45^\circ \sec^2 30^\circ)\)
  12. In right triangle ABC, \(\angle B = 90^\circ\), AB = 3 cm and AC = 6 cm. Find \(\angle C\) and \(\angle A\).
  13. If \(A = 30^\circ\), verify that: (i) \(\sin 2A = \frac{2 \tan A}{1 + \tan^2 A}\) (ii) \(\cos 2A = \frac{1 – \tan^2 A}{1 + \tan^2 A}\) (iii) \(\tan 2A = \frac{2 \tan A}{1 – \tan^2 A}\)
  14. If \(A = 45^\circ\), verify that (i) \(\sin 2A = 2\sin A \cos A\) (ii) \(\cos 2A = 2\cos^2 A – 1 = 1 – 2\sin^2 A\)
  15. Using the formula \(\cos A = \sqrt{\frac{1 + \cos 2A}{2}}\), find the value of \(\cos 30^\circ\), given \(\cos 60^\circ = 1/2\).
  16. Using the formula \(\sin A = \sqrt{\frac{1 – \cos 2A}{2}}\), find the value of \(\sin 30^\circ\), given \(\cos 60^\circ = 1/2\).
  17. Using the formula \(\tan 2A = \frac{2 \tan A}{1 – \tan^2 A}\), find the value of \(\tan 60^\circ\), given \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).
  18. If \(\sin (A – B) = \frac{1}{2}\) and \(\cos(A + B) = \frac{1}{2}\), then find the value of A and B.
  19. If \(\sin (A + B) = 1\) and \(\cos(A – B) = 1\), then find the value of A and B.
  20. If \(\tan (A – B) = \frac{1}{\sqrt{3}}\) and \(\tan (A + B) = \sqrt{3}\), then find the value of A and B.
  21. If \(\cos (A – B) = \frac{\sqrt{3}}{2}\) and \(\sin (A + B) = 1\), then find the value of A and B.
  22. If A and B are acute angles such that \(\tan A = \frac{1}{3}\), \(\tan B = \frac{1}{2}\) and \(\tan (A+B) = \frac{\tan A + \tan B}{1 – \tan A \tan B}\), show that \(A + B = 45^\circ\).
  23. If \(A = B = 45^\circ\), verify various identities like \(\sin(A+B)\), etc.
  24. If \(A = 60^\circ\) and \(B = 30^\circ\), verify various identities like \(\sin(A+B)\), etc.
  25. Evaluate: \(\frac{3\cos^2 60^\circ + 2\cot^2 30^\circ – 5\sin^2 45^\circ}{\sec^2 60^\circ – \text{cosec}^2 45^\circ}\) (Note: Typo in source, interpreted).

PRACTICE QUESTIONS: COMPLEMENTARY ANGLES

  1. Evaluate: \(\cot \theta \tan(90^\circ – \theta) – \sec (90^\circ – \theta) \text{cosec } \theta + (\sin^2 25^\circ + \sin^2 65^\circ) + \sqrt{3} (\tan 5^\circ \tan 15^\circ \tan 30^\circ \tan 75^\circ \tan 85^\circ)\).
  2. Evaluate without using tables: \(\frac{\sec \theta \text{cosec } (90^\circ – \theta) – \tan \theta \cot(90^\circ – \theta) + \sin^2 55^\circ + \sin^2 35^\circ}{\tan 10^\circ \tan 20^\circ \tan 60^\circ \tan 70^\circ \tan 80^\circ}\).
  3. Evaluate: \(\frac{\sec^2 54^\circ – \cot^2 36^\circ}{\text{cosec}^2 57^\circ – \tan^2 33^\circ} + 2\sin^2 38^\circ \sec^2 52^\circ – \sin^2 45^\circ\).
  4. Express \(\sin 67^\circ + \cos 75^\circ\) in terms of trigonometric ratios of angles between \(0^\circ\) and \(45^\circ\).
  5. If \(\sin 4A = \cos(A – 20^\circ)\), where A is an acute angle, find the value of A.
  6. If A, B and C are the interior angles of triangle ABC, prove that \(\tan\left(\frac{B+C}{2}\right) = \cot\left(\frac{A}{2}\right)\).
  7. If A, B, C are interior angles of a \(\Delta ABC\), then show that \(\cos\left(\frac{B+C}{2}\right) = \sin\left(\frac{A}{2}\right)\).
  8. If A, B, C are interior angles of a \(\Delta ABC\), then show that \(\cos\left(\frac{B+C}{2}\right) = \sec\left(\frac{A}{2}\right)\). (Note: Likely typo in source for RHS, usually sine/cosine pair).
  9. If A, B, C are interior angles of a \(\Delta ABC\), then show that \(\cot\left(\frac{C+A}{2}\right) = \tan\left(\frac{B}{2}\right)\).
  10. Without using trigonometric tables, find the value of \(\frac{\cos 70^\circ}{\sin 20^\circ} + \cos 57^\circ \text{cosec } 33^\circ – 2\cos 60^\circ\).
  11. If \(\sec 4A = \text{cosec }(A – 20^\circ)\), where 4A is an acute angle, find the value of A.
  12. If \(\tan 2A = \cot (A – 40^\circ)\), where 2A is an acute angle, find the value of A.
  13. Evaluate \(\tan 10^\circ \tan 15^\circ \tan 75^\circ \tan 80^\circ\).
  14. Evaluate: \(\left[\frac{\sin^2 22^\circ + \sin^2 68^\circ}{\cos^2 22^\circ + \cos^2 68^\circ} + \sin^2 63^\circ + \cos 63^\circ \sin 27^\circ\right]\).
  15. Express \(\tan 60^\circ + \cos 46^\circ\) in terms of trigonometric ratios of angles between \(0^\circ\) and \(45^\circ\).
  16. Express \(\sec 51^\circ + \text{cosec } 25^\circ\) in terms of trigonometric ratios of angles between \(0^\circ\) and \(45^\circ\).
  17. Express \(\cot 77^\circ + \sin 54^\circ\) in terms of trigonometric ratios of angles between \(0^\circ\) and \(45^\circ\).
  18. If \(\tan 3A = \cot (3A – 60^\circ)\), where 3A is an acute angle, find the value of A.
  19. If \(\sin 2A = \cos(A + 36^\circ)\), where 2A is an acute angle, find the value of A.
  20. If \(\text{cosec } A = \sec(A – 10^\circ)\), where A is an acute angle, find the value of A.
  21. If \(\sin 5\theta = \cos 4\theta\), where \(5\theta\) and \(4\theta\) are acute angles, find the value of \(\theta\).
  22. If \(\tan 2A = \cot (A – 18^\circ)\), where 2A is an acute angle, find the value of A.
  23. If \(\tan 2\theta = \cot(\theta + 60^\circ)\), where \(2\theta\) and \(\theta + 60^\circ\) are acute angles, find the value of \(\theta\).
  24. Evaluate: \(\frac{2\sin 68^\circ}{\cos 22^\circ} – \frac{2\cot 15^\circ}{5 \tan 75^\circ} – \frac{3 \tan 45^\circ \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ}{5}\).
  25. Evaluate: \(\frac{\cos(90^\circ – \theta) \sec(90^\circ – \theta) \tan \theta}{\text{cosec } (90^\circ – \theta) \sin(90^\circ – \theta) \cot(90^\circ – \theta)} + \frac{\tan(90^\circ – \theta)}{\cot \theta} – 2\).
  26. Evaluate: \(\frac{\sin 18^\circ}{\cos 72^\circ} + \sqrt{3} \{\tan 10^\circ \tan 30^\circ \tan 40^\circ \tan 50^\circ \tan 80^\circ\}\).
  27. Evaluate: \(\frac{3\cos 55^\circ}{7\sin 35^\circ} – \frac{4(\cos 70^\circ \text{cosec } 20^\circ)}{7(\tan 5^\circ \tan 25^\circ \tan 45^\circ \tan 65^\circ \tan 85^\circ)}\).
  28. Evaluate: \(\cos(40^\circ – \theta) – \sin(50^\circ + \theta) + \frac{\cos^2 40^\circ + \cos^2 50^\circ}{\sin^2 40^\circ + \sin^2 50^\circ}\).
  29. If \(A + B = 90^\circ\), prove that \(\sqrt{\frac{\tan A \tan B + \tan A \cot B}{\sin A \sec B} – \frac{\sin^2 B}{\cos^2 A}} = \tan A\).
  30. If \(\cos 2\theta = \sin 4\theta\), where \(2\theta\) and \(4\theta\) are acute angles, find the value of \(\theta\).

PRACTICE QUESTIONS: TRIGONOMETRIC IDENTITIES

  1. Prove that \(\frac{\cos \theta}{1 – \tan \theta} + \frac{\sin \theta}{1 – \cot \theta} = \sin \theta + \cos \theta\).
  2. Prove that \(\frac{1}{2} \left\{ \frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} \right\} = \frac{1}{\sin \theta}\).
  3. Prove that: \(\frac{\tan^3 \alpha}{1 + \tan^2 \alpha} + \frac{\cot^3 \alpha}{1 + \cot^2 \alpha} = \sec \alpha \text{cosec } \alpha – 2\sin \alpha \cos \alpha\).
  4. Prove that: \(\frac{\tan A}{1 – \cot A} + \frac{\cot A}{1 – \tan A} = 1 + \sec A \text{cosec } A\).
  5. Prove that: \(\frac{\sqrt{1 + \sin A}}{\sqrt{1 – \sin A}} + \frac{\sqrt{1 – \sin A}}{\sqrt{1 + \sin A}} = 2 \sec A\). (Assuming similar structure to standard problems).
  6. Prove that \((\tan A + \text{cosec } B)^2 – (\cot B – \sec A)^2 = 2\tan A \cot B (\text{cosec } A + \sec B)\).
  7. Prove that: \(\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A\).
  8. Prove that: \(\frac{1 – \cos A + \sin A}{\sin A + \cos A – 1} = \frac{1 + \cos A}{\sin A}\) (Standard Identity format).
  9. Prove that: \(\frac{\sin^2 A}{\cos^2 A} + \frac{\cos^2 A}{\sin^2 A} = \frac{1}{\sin^2 A \cos^2 A} – 2\).
  10. Prove that \(\frac{\cos A}{1 – \tan A} – \frac{\sin^2 A}{\cos A – \sin A} = \cos A + \sin A\).
  11. Prove that \(\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1 – \cos A}\).
  12. Prove that: \(\frac{\tan A + \sec A – 1}{\tan A – \sec A + 1} = \frac{1 + \sin A}{\cos A}\).
  13. Prove that: \((\tan \theta + \sec \theta – 1) \cdot (\tan \theta + 1 + \sec \theta) = \frac{2\sin \theta}{1 – \sin \theta}\) (Check signs based on standard identities).
  14. If \(x = a \sin \theta + b \cos \theta\) and \(y = a \cos \theta – b \sin \theta\), prove that \(x^2 + y^2 = a^2 + b^2\).
  15. If \(\sec \theta + \tan \theta = m\), show that \(\frac{m^2 – 1}{m^2 + 1} = \sin \theta\).
  16. Prove that: \(\frac{1 + \cos \theta + \sin \theta}{1 + \cos \theta – \sin \theta} = \frac{1 + \sin \theta}{\cos \theta}\).
  17. Prove that \(\sec^4 A (1 – \sin^4 A) – 2 \tan^2 A = 1\).
  18. If \(\text{cosec } \theta – \sin \theta = m\) and \(\sec \theta – \cos \theta = n\), prove that \((m^2 n)^{2/3} + (mn^2)^{2/3} = 1\).
  19. If \(\tan \theta + \sin \theta = m\) and \(\tan \theta – \sin \theta = n\), show that \(m^2 – n^2 = 4\sqrt{mn}\).
  20. If \(a \cos \theta – b \sin \theta = c\), prove that \(a \sin \theta + b \cos \theta = \pm \sqrt{a^2 + b^2 – c^2}\).
  21. If \(\cos \theta + \sin \theta = \sqrt{2} \cos \theta\), prove that \(\cos \theta – \sin \theta = \sqrt{2} \sin \theta\).
  22. If \(\frac{x}{a} \sin \theta – \frac{y}{b} \cos \theta = 1\) and \(\frac{x}{a} \cos \theta + \frac{y}{b} \sin \theta = 1\), prove that \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 2\).
  23. If \(\tan \theta + \sin \theta = m\) and \(\tan \theta – \sin \theta = n\), prove that \(m^2 – n^2 = 4\sqrt{mn}\) (Note: Duplicate of Q19 with formula error check).
  24. If \(\text{cosec } \theta – \sin \theta = a^3\) and \(\sec \theta – \cos \theta = b^3\), prove that \(a^2 b^2 (a^2 + b^2) = 1\).
  25. If \(x = a \cos^3 \theta\) and \(y = a \sin^3 \theta\), prove that \((x/a)^{2/3} + (y/a)^{2/3} = 1\).
  26. Prove that \(\text{cosec}^2 \theta + \sec^2 \theta = \text{cosec}^2 \theta \sec^2 \theta\).
  27. Prove the identity: \(\sqrt{\frac{1 + \sin \theta}{1 – \sin \theta}} + \sqrt{\frac{1 – \sin \theta}{1 + \sin \theta}} = 2 \sec \theta\).
  28. Prove the identity: \(\sec^6 \theta = \tan^6 \theta + 3 \tan^2 \theta \sec^2 \theta + 1\).
  29. Prove the identity: \((\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A\).
  30. If \(x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta\) and \(x \sin \theta = y \cos \theta\), prove that \(x^2 + y^2 = 1\).
  31. If \(\sec \theta = x + \frac{1}{4x}\), Prove that \(\sec \theta + \tan \theta = 2x\) or \(\frac{1}{2x}\).
  32. Prove that \(\left( \frac{1 + \tan^2 A}{1 + \cot^2 A} \right) = \left( \frac{1 – \tan A}{1 – \cot A} \right)^2 = \tan^2 A\).
  33. If \(\cot \theta + \tan \theta = x\) and \(\sec \theta – \cos \theta = y\), prove that \((x^2 y)^{2/3} – (xy^2)^{2/3} = 1\).
  34. If \(\frac{\cos \alpha}{\cos \beta} = m\) and \(\frac{\cos \alpha}{\sin \beta} = n\), show that \((m^2 + n^2) \cos^2 \beta = n^2\).
  35. If \(\text{cosec } \theta – \sin \theta = a\) and \(\sec \theta – \cos \theta = b\), prove that \(a^2 b^2 (a^2 + b^2 + 3) = 1\).
  36. If \(x = r \sin A \cos C\), \(y = r \sin A \sin C\) and \(z = r \cos A\), prove that \(r^2 = x^2 + y^2 + z^2\).
  37. If \(\tan A = n \tan B\) and \(\sin A = m \sin B\), prove that \(\cos^2 A = \frac{m^2 – 1}{n^2 – 1}\).
  38. If \(\sin \theta + \sin^2 \theta = 1\), find the value of \(\cos^{12} \theta + 3\cos^{10} \theta + 3\cos^8 \theta + \cos^6 \theta + 2\cos^4 \theta + 2\cos^2 \theta – 2\).
  39. Prove that: \((1 – \sin \theta + \cos \theta)^2 = 2(1 + \cos \theta)(1 – \sin \theta)\).
  40. If \(\sin \theta + \sin^2 \theta = 1\), prove that \(\cos^2 \theta + \cos^4 \theta = 1\).
  41. If \(a \sec \theta + b \tan \theta + c = 0\) and \(p \sec \theta + q \tan \theta + r = 0\), prove that \((br – qc)^2 – (pc – ar)^2 = (aq – bp)^2\).
  42. If \(\sin \theta + \sin^2 \theta + \sin^3 \theta = 1\), then prove that \(\cos^6 \theta – 4\cos^4 \theta + 8\cos^2 \theta = 4\).
  43. If \(\tan^2 \theta = 1 – a^2\), prove that \(\sec \theta + \tan^3 \theta \text{cosec } \theta = (2 – a^2)^{3/2}\).
  44. If \(x = a \sec \theta + b \tan \theta\) and \(y = a \tan \theta + b \sec \theta\), prove that \(x^2 – y^2 = a^2 – b^2\).
  45. If \(3 \sin \theta + 5 \cos \theta = 5\), prove that \(5 \sin \theta – 3 \cos \theta = \pm 3\).

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