✦ Exam Vault: Chapter-Wise PYQs (2015-2026) | Chapter 8: Introduction of trigonometry

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✦ NCF 2023 ALIGNED • CBSE CLASS X ✦

Chapter 8: Introduction to Trigonometry

CBSE Official PYQs (2015-2026) • Mega Set

✨ Explore Interactive TLMs, Worksheets & Visual Notes at: www.akashmaths.online

📅 CBSE 2026 (30/2/1)⭐ 1 Mark📝 MCQ

Q1. If \(\cos A = \frac{4}{5}\), then the value of \(\tan A\) is :

(A) \(\frac{3}{5}\)

(B) \(\frac{3}{4}\)

(C) \(\frac{4}{3}\)

(D) \(\frac{5}{3}\)

Correct Answer: (B) \(\frac{3}{4}\)

📅 CBSE 2026 (30/2/1)⭐ 1 Mark📝 MCQ

Q2. If \(2 \sin A = 1\), then the value of \(\tan A + \cot A\) is :

(A) \(\sqrt{3}\)

(B) \(\frac{4}{\sqrt{3}}\)

(C) \(\frac{\sqrt{3}}{2}\)

(D) \(1\)

Correct Answer: (B) \(\frac{4}{\sqrt{3}}\)

📅 CBSE 2024 (30/1/2)⭐ 1 Mark📝 MCQ

Q3. If \(2 \tan A = 3\), then the value of \(\frac{4 \sin A + 3 \cos A}{4 \ \sin A - 3 \cos A}\) is :

(A) \(\frac{7}{\sqrt{13}}\)

(B) \(\frac{1}{\sqrt{13}}\)

(C) \(3\)

(D) does not exist

Correct Answer: (C) \(3\)

📅 CBSE 2023 (30/3/1)⭐ 1 Mark📝 MCQ

Q4. If \(\tan 3\theta = \sqrt{3}\), then the value of \(\frac{\theta}{2}\) is :

(A) \(60^\circ\)

(B) \(30^\circ\)

(C) \(20^\circ\)

(D) \(10^\circ\)

Correct Answer: (D) \(10^\circ\)

📅 CBSE 2023 (30/2/3)⭐ 1 Mark📝 MCQ

Q5. If \(\sin \theta = \cos \theta\) (\(0^\circ < \theta < 90^\circ\)), then the value of \(\sec \theta \cdot \sin \theta\) is :

(A) \(\frac{1}{\sqrt{2}}\)

(B) \(\sqrt{2}\)

(C) \(0\)

(D) \(1\)

Correct Answer: (D) \(1\)

📅 CBSE 2020 (30/1/1)⭐ 1 Mark📝 MCQ

Q6. If \(\sin A + \sin^2 A = 1\), then the value of the expression \((\cos^2 A + \cos^4 A)\) is :

(A) \(1\)

(B) \(\frac{1}{2}\)

(C) \(2\)

(D) \(3\)

Correct Answer: (A) \(1\)

📅 CBSE 2025 (30/1/1)⭐ 1 Mark📝 MCQ

Q7. The value of \((\sin^2 30^\circ + \cos^2 30^\circ)\) is :

(A) \(0\)

(B) \(1\)

(C) \(2\)

(D) \(\frac{1}{2}\)

Correct Answer: (B) \(1\)

📅 CBSE 2024 (30/1/1)⭐ 1 Mark📝 MCQ

Q8. If \(x \tan 45^\circ \cos 60^\circ = \sin 60^\circ \cot 60^\circ\), then \(x\) is equal to :

(A) \(1\)

(B) \(\frac{1}{2}\)

(C) \(\frac{1}{\sqrt{2}}\)

(D) \(\sqrt{3}\)

Correct Answer: (A) \(1\)

📅 CBSE 2023 (30/3/2)⭐ 1 Mark📝 MCQ

Q9. If \(\sin \theta + \cos \theta = \sqrt{2} \cos \theta\), (\(\theta \neq 90^\circ\)) then the value of \(\tan \theta\) is :

(A) \(\sqrt{2} - 1\)

(B) \(\sqrt{2} + 1\)

(C) \(\frac{1}{\sqrt{2}}\)

(D) \(\sqrt{3}\)

Correct Answer: (A) \(\sqrt{2} - 1\)

📅 CBSE 2020 (30/1/2)⭐ 1 Mark📝 MCQ

Q10. If \(\sin A = \frac{1}{2}\), then the value of \(\cot A\) is :

(A) \(\sqrt{3}\)

(B) \(\frac{1}{\sqrt{3}}\)

(C) \(\frac{\sqrt{3}}{2}\)

(D) \(1\)

Correct Answer: (A) \(\sqrt{3}\)

📅 CBSE 2025 (30/1/3)⭐ 1 Mark📝 MCQ

Q11. If \(\tan \theta = \frac{a}{b}\), then the value of \(\frac{a \sin \theta - b \cos \theta}{a \sin \theta + b \cos \theta}\) is :

(A) \(\frac{a^2+b^2}{a^2-b^2}\)

(B) \(\frac{a^2-b^2}{a^2+b^2}\)

(C) \(\frac{a}{\sqrt{a^2+b^2}}\)

(D) \(\frac{b}{\sqrt{a^2+b^2}}\)

Correct Answer: (B) \(\frac{a^2-b^2}{a^2+b^2}\)

📅 CBSE 2026 (30/2/2)⭐ 1 Mark📝 MCQ

Q12. If \(\sec \theta + \tan \theta = p\), then the value of \(\sec \theta - \tan \theta\) is :

(A) \(p\)

(B) \(\frac{1}{p}\)

(C) \(-p\)

(D) \(p^2\)

Correct Answer: (B) \(\frac{1}{p}\)

📅 CBSE 2019 (30/1/1)⭐ 1 Mark📝 MCQ

Q13. Value of \(\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ}\) is :

(A) \(\sin 60^\circ\)

(B) \(\cos 60^\circ\)

(C) \(\tan 60^\circ\)

(D) \(\sin 30^\circ\)

Correct Answer: (A) \(\sin 60^\circ\)

📅 CBSE 2024 (30/2/1)⭐ 1 Mark📝 MCQ

Q14. In \(\Delta ABC\), right-angled at \(B\), if \(\tan A = \frac{1}{\sqrt{3}}\), then the value of \(\sin A \cos C + \cos A \sin C\) is :

(A) \(0\)

(B) \(1\)

(C) \(\frac{1}{2}\)

(D) \(\frac{\sqrt{3}}{2}\)

Correct Answer: (B) \(1\)

📅 CBSE 2023 (30/3/1)⭐ 1 Mark📝 MCQ

Q15. If \(\sqrt{3} \sin \theta - \cos \theta = 0\) (\(0^\circ < \theta \le 90^\circ\)), then the value of \(\theta\) is :

(A) \(30^\circ\)

(B) \(45^\circ\)

(C) \(60^\circ\)

(D) \(90^\circ\)

Correct Answer: (A) \(30^\circ\)

📅 CBSE 2025 (30/1/2)⭐ 1 Mark📝 MCQ

Q16. The value of \((\sec^2 \theta - 1)(\cot^2 \theta)\) is :

(A) \(0\)

(B) \(1\)

(C) \(-1\)

(D) \(2\)

Correct Answer: (B) \(1\)

📅 CBSE 2020 (30/1/1)⭐ 1 Mark📝 MCQ

Q17. If \(\sin \alpha = \frac{1}{2}\) and \(\cos \beta = \frac{1}{2}\), then the value of \((\alpha + \beta)\) is :

(A) \(0^\circ\)

(B) \(30^\circ\)

(C) \(60^\circ\)

(D) \(90^\circ\)

Correct Answer: (D) \(90^\circ\)

📅 CBSE 2018 (30/1)⭐ 1 Mark📝 MCQ

Q18. If \(4 \tan \theta = 3\), then \(\frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta}\) is equal to :

(A) \(\frac{2}{3}\)

(B) \(\frac{1}{3}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{3}{4}\)

Correct Answer: (C) \(\frac{1}{2}\)

📅 CBSE 2026 (30/2/3)⭐ 1 Mark📝 MCQ

Q19. If \(x = a \cos \theta\) and \(y = b \sin \theta\), then \(b^2 x^2 + a^2 y^2\) is equal to :

(A) \(a^2 b^2\)

(B) \(a b\)

(C) \(a^2 + b^2\)

(D) \(a^4 b^4\)

Correct Answer: (A) \(a^2 b^2\)

📅 CBSE 2019 (30/1/2)⭐ 1 Mark📝 MCQ

Q20. If \(\tan \theta + \cot \theta = 2\), then the value of \(\tan^2 \theta + \cot^2 \theta\) is :

(A) \(2\)

(B) \(4\)

(C) \(1\)

(D) \(5\)

Correct Answer: (A) \(2\)

📅 CBSE 2020 (30/1/3)⭐ 1 Mark📝 MCQ

Q21. If \(\sin \theta = \frac{a}{b}\), then \(\cos \theta\) is equal to :

(A) \(\frac{b}{\sqrt{b^2-a^2}}\)

(B) \(\frac{\sqrt{b^2-a^2}}{b}\)

(C) \(\frac{\sqrt{b^2-a^2}}{a}\)

(D) \(\frac{a}{\sqrt{b^2-a^2}}\)

Correct Answer: (B) \(\frac{\sqrt{b^2-a^2}}{b}\)

📅 CBSE 2024 (30/1/3)⭐ 1 Mark📝 MCQ

Q22. Given that \(\sin \alpha = \frac{\sqrt{3}}{2}\) and \(\cos \beta = 0\), then the value of \(\beta - \alpha\) is :

(A) \(0^\circ\)

(B) \(90^\circ\)

(C) \(60^\circ\)

(D) \(30^\circ\)

Correct Answer: (D) \(30^\circ\)

📅 CBSE 2025 (30/1/1)⭐ 1 Mark📝 MCQ

Q23. The value of \((1 + \tan^2 \theta)(1 - \sin \theta)(1 + \sin \theta)\) is :

(A) \(0\)

(B) \(1\)

(C) \(-1\)

(D) \(2\)

Correct Answer: (B) \(1\)

📅 CBSE 2017 (30/1)⭐ 1 Mark📝 MCQ

Q24. If \(\cot \theta = \frac{7}{8}\), then the value of \(\frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}\) is :

(A) \(\frac{7}{8}\)

(B) \(\frac{64}{49}\)

(C) \(\frac{49}{64}\)

(D) \(\frac{8}{7}\)

Correct Answer: (C) \(\frac{49}{64}\)

📅 CBSE 2026 (30/2/1)⭐ 1 Mark📝 MCQ

Q25. If \(\sec A = \frac{15}{7}\) and \(A+B = 90^\circ\), then the value of \(\csc B\) is :

(A) \(\frac{7}{15}\)

(B) \(\frac{15}{7}\)

(C) \(\frac{15}{8}\)

(D) \(\frac{8}{15}\)

Correct Answer: (B) \(\frac{15}{7}\)

📅 CBSE 2018 (30/2)⭐ 1 Mark📝 MCQ

Q26. If \(\sin \theta - \cos \theta = 0\), then the value of \((\sin^4 \theta + \cos^4 \theta)\) is :

(A) \(1\)

(B) \(\frac{3}{4}\)

(C) \(\frac{1}{2}\)

(D) \(\frac{1}{4}\)

Correct Answer: (C) \(\frac{1}{2}\)

📅 CBSE 2016 (30/1)⭐ 1 Mark📝 MCQ

Q27. If \(\sin \theta + \sin^2 \theta = 1\), then the value of \((\cos^{12} \theta + 3 \cos^{10} \theta + 3 \cos^8 \theta + \cos^6 \theta - 1)\) is :

(A) \(0\)

(B) \(1\)

(C) \(-1\)

(D) \(2\)

Correct Answer: (A) \(0\)

📅 CBSE 2020 (30/1/1)⭐ 1 Mark📝 MCQ

Q28. Value of \((\tan 1^\circ \tan 2^\circ \tan 3^\circ \dots \tan 89^\circ)\) is :

(A) \(0\)

(B) \(1\)

(C) \(-1\)

(D) \(\frac{1}{2}\)

Correct Answer: (B) \(1\)

📅 CBSE 2025 (30/1/3)⭐ 1 Mark📝 MCQ

Q29. If \(3 \cos \theta = 1\), then the value of \(\csc \theta\) is :

(A) \(\frac{3}{2\sqrt{2}}\)

(B) \(\frac{2\sqrt{2}}{3}\)

(C) \(3\)

(D) \(\frac{1}{3}\)

Correct Answer: (A) \(\frac{3}{2\sqrt{2}}\)

📅 CBSE 2024 (30/1/1)⭐ 1 Mark📝 MCQ

Q30. Assertion (A): The value of \(\sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ\) is \(1\).
Reason (R): For any acute angle \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\).

(A) Both (A) and (R) are true and (R) is correct explanation.

(B) Both (A) and (R) are true but (R) is not correct explanation.

(C) (A) is true but (R) is false.

(D) (A) is false but (R) is true.

Correct Answer: (A) Both (A) and (R) are true and (R) is the correct explanation of (A).

📅 CBSE 2023 (30/3/2)⭐ 1 Mark📝 MCQ

Q31. If \(\tan \theta + \sec \theta = l\), then \(\sec \theta\) is equal to :

(A) \(\frac{l^2-1}{2l}\)

(B) \(\frac{l^2+1}{2l}\)

(C) \(\frac{l^2+1}{l}\)

(D) \(\frac{l^2-1}{l}\)

Correct Answer: (B) \(\frac{l^2+1}{2l}\)

📅 CBSE 2026 (30/2/3)⭐ 1 Mark📝 MCQ

Q32. If \(\sec \theta = \frac{17}{8}\), then the value of \(\cot \theta\) is :

(A) \(\frac{15}{8}\)

(B) \(\frac{8}{15}\)

(C) \(\frac{17}{15}\)

(D) \(\frac{15}{17}\)

Correct Answer: (B) \(\frac{8}{15}\)

📅 CBSE 2019 (30/1/3)⭐ 1 Mark📝 MCQ

Q33. If \(5 \tan \theta = 4\), then the value of \(\frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta}\) is :

(A) \(\frac{1}{6}\)

(B) \(\frac{1}{3}\)

(C) \(\frac{2}{5}\)

(D) \(\frac{3}{5}\)

Correct Answer: (A) \(\frac{1}{6}\)

📅 CBSE 2024 (30/1/2)⭐ 1 Mark📝 MCQ

Q34. If \(\cos (A+B) = 0\) and \(\sin (A-B) = \frac{1}{2}\), then the values of \(A\) and \(B\) are :

(A) \(A = 60^\circ, B = 30^\circ\)

(B) \(A = 45^\circ, B = 45^\circ\)

(C) \(A = 30^\circ, B = 60^\circ\)

(D) \(A = 90^\circ, B = 0^\circ\)

Correct Answer: (A) \(A = 60^\circ, B = 30^\circ\)

📅 CBSE 2015 (30/1)⭐ 1 Mark📝 MCQ

Q35. If \(\sec^2 \theta (1 + \sin \theta)(1 - \sin \theta) = k\), then the value of \(k\) is :

(A) \(0\)

(B) \(1\)

(C) \(2\)

(D) \(-1\)

Correct Answer: (B) \(1\)

📅 CBSE 2020 (30/1/1)⭐ 1 Mark📝 MCQ

Q36. If \(\sin \ \theta = \frac{1}{3}\), then the value of \((2 \cot^2 \theta + 2)\) is :

(A) \(18\)

(B) \(9\)

(C) \(16\)

(D) \(2\)

Correct Answer: (A) \(18\)

📅 CBSE 2025 (30/1/2)⭐ 1 Mark📝 MCQ

Q37. If \(4 \cos \theta = 11 \sin \theta\), then the value of \(\frac{11 \cos \theta - 7 \sin \theta}{11 \cos \theta + 7 \sin \theta}\) is :

(A) \(\frac{93}{149}\)

(B) \(\frac{90}{137}\)

(C) \(\frac{93}{137}\)

(D) \(\frac{11}{18}\)

Correct Answer: (C) \(\frac{93}{137}\)

📅 CBSE 2026 (30/2/2)⭐ 1 Mark📝 MCQ

Q38. Assertion (A): If \(\cos A + \cos^2 A = 1\), then \(\sin^2 A + \sin^4 A = 1\).
Reason (R): For any angle \(A\), \(\cos^2 A = 1 - \sin^2 A\).

(A) Both (A) and (R) are true and (R) is correct explanation.

(B) Both (A) and (R) are true but (R) is not correct explanation.

(C) (A) is true but (R) is false.

(D) (A) is false but (R) is true.

Correct Answer: (A) Both (A) and (R) are true and (R) is the correct explanation of (A).

📅 CBSE 2023 (30/2/3)⭐ 1 Mark📝 MCQ

Q39. The value of \((\csc^2 45^\circ - \cot^2 45^\circ) \cdot \sin^2 90^\circ\) is :

(A) \(0\)

(B) \(1\)

(C) \(2\)

(D) \(-1\)

Correct Answer: (B) \(1\)

📅 CBSE 2018 (30/1)⭐ 1 Mark📝 MCQ

Q40. If \(\sin \theta = \cos \theta\), then the value of \(2 \tan^2 \theta + \cos^2 \theta\) is :

(A) \(2\)

(B) \(\frac{5}{2}\)

(C) \(\frac{3}{2}\)

(D) \(\frac{1}{2}\)

Correct Answer: (B) \(\frac{5}{2}\)

📅 CBSE 2025 (30/1/1)⭐ 2 Marks📝 SA-I

Q41. If \(\sin (A - B) = \frac{1}{2}\) and \(\cos (A + B) = \frac{1}{2}\) where \(0^\circ < A + B \le 90^\circ\) and \(A > B\), find the values of \(A\) and \(B\).

Solution: \(A = 45^\circ\), \(B = 15^\circ\)

📅 CBSE 2023 (30/3/1)⭐ 2 Marks📝 SA-I

Q42. Evaluate: \(\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}\).

Solution: \(\frac{67}{12}\)

📅 CBSE 2024 (30/1/2)⭐ 2 Marks📝 SA-I

Q43. Prove that: \(\frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A\).

Solution: Proof: \(\frac{1+\tan^2 A}{1+\cot^2 A} = \frac{\sec^2 A}{\csc^2 A} = \frac{1/\cos^2 A}{1/\sin^2 A} = \tan^2 A\)

📅 CBSE 2018 (30/1)⭐ 2 Marks📝 SA-I

Q44. If \(\sqrt{3} \tan \theta = 3 \sin \theta\), find the value of \(\sin^2 \theta - \cos^2 \theta\).

Solution: \(\frac{1}{3}\)

📅 CBSE 2019 (30/1/1)⭐ 2 Marks📝 SA-I

Q45. Prove that: \(\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A\).

Solution: Proof by rationalization inside root.

📅 CBSE 2020 (30/1/2)⭐ 2 Marks📝 SA-I

Q46. If \(15 \cot A = 8\), find the values of \(\sin A\) and \(\sec A\).

Solution: \(\sin A = \frac{15}{17}\), \(\sec A = \frac{17}{8}\)

📅 CBSE 2026 (30/2/1)⭐ 2 Marks📝 SA-I

Q47. Evaluate: \(\frac{4}{\cot^2 30^\circ} + \frac{1}{\sin^2 60^\circ} - 3 \cos^2 45^\circ\).

Solution: \(\frac{25}{12}\)

📅 CBSE 2017 (30/1)⭐ 2 Marks📝 SA-I

Q48. If \(\cos \theta + \sin \theta = \sqrt{2} \cos \theta\), prove that \(\cos \theta - \sin \theta = \sqrt{2} \sin \theta\).

Solution: Proof completed by algebra.

📅 CBSE 2016 (30/1)⭐ 2 Marks📝 SA-I

Q49. Prove that: \((\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = 7 + \tan^2 \theta + \cot^2 \theta\).

Solution: Proof completed by expanding squares.

📅 CBSE 2024 (30/1/1)⭐ 2 Marks📝 SA-I

Q50. If \(\tan (A + B) = \sqrt{3}\) and \(\tan (A - B) = \frac{1}{\sqrt{3}}\) (\(0^\circ < A + B \le 90^\circ\); \(A > B\)), find the measures of angles \(A\) and \(B\).

Solution: \(A = 45^\circ\), \(B = 15^\circ\)

📅 CBSE 2025 (30/1/3)⭐ 2 Marks📝 SA-I

Q51. Prove that: \(\frac{\cos A}{1 - \sin A} + \frac{\cos A}{1 + \sin A} = 2 \sec A\).

Solution: Proof completed by common denominator.

📅 CBSE 2023 (30/2/3)⭐ 2 Marks📝 SA-I

Q52. If \(\sin \theta + \cos \theta = p\) and \(\sec \theta + \csc \theta = q\), show that \(q(p^2 - 1) = 2p\).

Solution: Algebraic substitution and proof.

📅 CBSE 2026 (30/2/2)⭐ 2 Marks📝 SA-I

Q53. Find the value of \(\theta\) if \(\frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4\) (\(0^\circ < \theta < 90^\circ\)).

Solution: \(\theta = 60^\circ\)

📅 CBSE 2015 (30/1)⭐ 2 Marks📝 SA-I

Q54. Prove that: \(\frac{\tan \theta + \sin \theta}{\tan \theta - \sin \theta} = \frac{\sec \theta + 1}{\sec \theta - 1}\).

Solution: Proof by converting \(\tan\) to \(\sin/\cos\).

📅 CBSE 2018 (30/3)⭐ 2 Marks📝 SA-I

Q55. If \(3 \cot A = 4\), check whether \(\frac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A - \sin^2 A\) or not.

Solution: Yes, both sides are equal to \(\frac{7}{25}\).

📅 CBSE 2024 (30/1/1)⭐ 3 Marks📝 SA-II

Q56. Prove that: \(\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta\).

Solution: Proof by factoring out \(\sin \theta\) and \(\cos \theta\).

📅 CBSE 2023 (30/3/1)⭐ 3 Marks📝 SA-II

Q57. Prove that: \(\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A\) using the identity \(\csc^2 A = 1 + \cot^2 A\).

Solution: Proof by dividing numerator and denominator by \(\sin A\).

📅 CBSE 2020 (30/1/1)⭐ 3 Marks📝 SA-II

Q58. Prove that: \(\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \csc \theta\).

Solution: Proof by converting into \(\sin\) and \(\cos\).

📅 CBSE 2019 (30/1/2)⭐ 3 Marks📝 SA-II

Q59. If \(\sec \theta + \tan \theta = p\), prove that \(\sin \theta = \frac{p^2 - 1}{p^2 + 1}\).

Solution: Proof completed correctly.

📅 CBSE 2026 (30/2/1)⭐ 3 Marks📝 SA-II

Q60. Prove the following identity: \(\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} + \sin \theta \cos \theta = 1\).

Solution: Proof by using \(a^3+b^3\) expansion.

📅 CBSE 2018 (30/1)⭐ 3 Marks📝 SA-II

Q61. Prove that: \(\frac{\cot A - \cos A}{\cot A + \cos A} = \frac{\csc A - 1}{\csc A + 1}\).

Solution: Proof by converting \(\cot A\) to \(\cos A/\sin A\).

📅 CBSE 2025 (30/1/2)⭐ 3 Marks📝 SA-II

Q62. If \(\sin \theta + \cos \theta = \sqrt{3}\), then prove that \(\tan \theta + \cot \theta = 1\).

Solution: Proof by squaring both sides.

📅 CBSE 2024 (30/1/3)⭐ 3 Marks📝 SA-II

Q63. Prove that: \((\sec A - \tan A)^2 = \frac{1 - \sin A}{1 + \sin A}\).

Solution: Algebraic identity proof.

📅 CBSE 2016 (30/1)⭐ 3 Marks📝 SA-II

Q64. If \(x = p \sec \theta + q \tan \theta\) and \(y = p \tan \theta + q \sec \theta\), prove that \(x^2 - y^2 = p^2 - q^2\).

Solution: Proof by squaring and subtracting.

📅 CBSE 2025 (30/1/1)⭐ 3 Marks📝 SA-II

Q65. Prove that: \(\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} = 2 \csc \theta\).

Solution: Proof by cross multiplying.

📅 CBSE 2023 (30/3/2)⭐ 3 Marks📝 SA-II

Q66. Prove that: \(\frac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \frac{1 + \sin A}{\cos A}\).

Solution: Proof by using \(1 = \sec^2 A - \tan^2 A\).

📅 CBSE 2026 (30/2/3)⭐ 3 Marks📝 SA-II

Q67. If \(\sin \theta + 2 \cos \theta = 1\), then prove that \(2 \sin \theta - \cos \theta = 2\).

Solution: Proof by squaring and swapping variables.

📅 CBSE 2025 (30/1/1)⭐ 5 Marks📝 LA

Q68. Prove the following identity: \(\frac{1}{\csc A - \cot A} - \frac{1}{\sin A} = \frac{1}{\sin A} - \frac{1}{\csc A + \cot A}\).

Solution: Proof by showing both sides equal to \(\cot A\).

📅 CBSE 2024 (30/1/2)⭐ 5 Marks📝 LA

Q69. If \(\csc \theta + \cot \theta = q\), show that \(\csc \theta - \cot \theta = \frac{1}{q}\) and hence find the values of \(\cos \theta\) and \(\sin \theta\).

Solution: \(\cos \theta = \frac{q^2-1}{q^2+1}\), \(\sin \theta = \frac{2q}{q^2+1}\)

📅 CBSE 2019 (30/1/1)⭐ 5 Marks📝 LA

Q70. Prove that: \(\frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{\sin^2 A - \cos^2 A} = \frac{2}{2\sin^2 A - 1}\).

Solution: Proof by common denominator.

📅 CBSE 2023 (30/3/1)⭐ 5 Marks📝 LA

Q71. If \(a \cos \theta - b \sin \theta = c\), prove that \(a \sin \theta + b \cos \theta = \pm \sqrt{a^2 + b^2 - c^2}\).

Solution: Proof by squaring equations.

📅 CBSE 2026 (30/2/2)⭐ 5 Marks📝 LA

Q72. Prove the identity: \(\frac{\cos^3 A + \sin^3 A}{\cos A + \sin A} + \frac{\cos^3 A - \sin^3 A}{\cos A - \sin A} = 2\).

Solution: Proof by algebraic expansion.

📅 CBSE 2015 (30/1)⭐ 5 Marks📝 LA

Q73. If \(\tan \theta + \sin \theta = m\) and \(\tan \theta - \sin \theta = n\), show that \(m^2 - n^2 = 4 \sqrt{m n}\).

Solution: Proof verified using difference of squares.

📅 CBSE 2024 (30/2/1)⭐ 4 Marks📝 Case Study

Q74. Case Study 1: Clinometer in Action
To measure the heights of inaccessible objects like towers, chimneys, or monuments, a simple device called a Clinometer is used by surveyors. It consists of a graduated semi-circular protractor with a plumb-line suspended from its center. A student of Class 10 makes a homemade clinometer and uses it to study various objects in his locality.

Based on the concepts of trigonometric ratios and trigonometry properties:

1. If the student looks at the top of a multi-storey building at an angle of \(45^\circ\) through the clinometer, and the base distance of the student from the building is \(30\text{ m}\) (eye level is \(1.5\text{ m}\) above the ground), write the trigonometric equation to represent this scenario. (1 Mark)
2. Find the total height of the multi-storey building. (1 Mark)
3. If the student stands \(30\sqrt{3}\text{ m}\) away from the building, what will be the angle of elevation seen through the clinometer? (2 Marks)

Solution:
1. \(h - 1.5 = 30 \tan 45^\circ\)
2. Total height = \(31.5\text{ m}\)
3. Angle \(\theta = 30^\circ\)

📅 CBSE 2024 (30/2/3)⭐ 4 Marks📝 Case Study

Q75. Case Study 2: Physiotherapy Clinic Movement Modeling
In a physiotherapy clinic, a biomechanics engineer uses an advanced motion tracking system to model knee rehabilitation exercises. During a leg extension exercise, the knee position and movement are traced mathematically. The displacement path of the ankle relative to the hip joint is modeled using trigonometric coordinate systems. The joint angle \(\theta\) is tracked as the patient extends their leg.

Based on the clinic's trigonometric motion model:

1. Find the value of \(\sec \theta \cdot \cos \theta + \tan \theta \cdot \cot \theta\). (1 Mark)
2. If the extension angle \(\theta\) is measured to be \(60^\circ\), calculate the value of \(2\sin^2 \theta + 3\cos^2 \theta\). (1 Mark)
3. Show that \(\frac{1-\sin^2 \theta}{\cos^2 \theta} = \sin^2 45^\circ + \cos^2 45^\circ\) is an identity holds true for all acute rehab angles \(\theta\). (2 Marks)

Solution:
1. Value = \(2\)
2. Value = \(9/4\)
3. Proof: LHS = \(1\), RHS = \(1\).

Live Practice: Chapter 8 Introduction to Trigonometry

Q 1

MCQ Score: 0/0