Section A: Multiple Choice Questions (1 Mark Each)

1. If α + β = 90° and α = 2β, then cos²α + sin²β is equal to:
(a) 0 (b) 1/2 (c) 1 (d) 2

OR

1. If sin 4θ = √3/2, then θ/3 equals:
(a) 60° (b) 20° (c) 15° (d) 5°

2. If x (2tan 30° / (1 + tan² 30°)) = y (2tan 30° / (1 - tan² 30°)), then x : y equals:
(a) 1:1 (b) 1:2 (c) 2:1 (d) 4:1

3. Given sin θ = a/b, then (sec θ + tan θ) is equal to:
(a) √[(b+a)/(b-a)] (b) √[(b-a)/(b+a)] (c) (b+a)/(b-a) (d) (b-a)/(b+a)

4. If 3 cot θ = 4, then the value of (1 - tan² θ) / (1 + tan² θ) is equivalent to:
(a) sin² θ - cos² θ (b) cos² θ - sin² θ (c) 1 (d) 0

5. If sin θ = √3/2, the value of all other T-ratios can be found; specifically, cos θ is:
(a) 0 (b) 1/2 (c) 1 (d) √3/2

Section B: Fill in the Blanks (1 Mark Each)

6. The value of (sin 60° cos 30° + cos 60° sin 30°) is ________.

7. If tan A = 1 in a right ΔABC (right-angled at B), then the value of 2 sin A cos A is ________.

8. The identity (1 + tan² θ)(1 - sin² θ) simplifies to ________.

9. If cosec θ = 2, then the value of (cot θ + sin θ / (1 + cos θ)) is ________.

10. If sin θ + cos θ = √2, then the value of sin θ . cos θ is ________.

Section C: Assertion-Reasoning (1 Mark Each)

11. Assertion (A): sin² θ + sin θ = 2 is an identity.
Reason (R): An identity must be true for all values of the angle θ.

OR

11. Assertion (A): sec² θ - tan² θ = 1.
Reason (R): 1 + cot² θ = cosec² θ.

12. Assertion (A): If cos θ = 0.6, then 5 sin θ - 3 tan θ = 0.
Reason (R): sin² θ + cos² θ = 1.

13. Assertion (A): The value of sin 60° is √3/2.
Reason (R): In ΔABC, if ∠A = 30°, then sin A = BC/AC.

14. Assertion (A): If A = 30°, then sin 2A = 2 sin A cos A.
Reason (R): sin 60° = √3/2 and 2 sin 30° cos 30° = 2(1/2)(√3/2).

15. Assertion (A): tan A + sin A = m and tan A - sin A = n implies m² - n² = 4√mn.
Reason (R): (m+n)(m-n) = 4 tan A sin A.

Section D: Very Short Answer Questions (2 Marks Each)

16. If x = a sin θ and y = b cos θ, find the value of b²x² + a²y².

OR

16. If sin θ + cos θ = √2, then find the value of sin θ . cos θ.

17. If 15 cot A = 8, find the values of sin A and sec A.

18. Evaluate: (5 cos² 60° + 4 sec² 30° - tan² 45°) / (sin² 30° + cos² 30°).

19. In ΔABC, right-angled at B, AB = 24 cm, BC = 7 cm. Find sin A and cos A.

20. If sin(A + B) = 1 and cos(A - B) = 1, find A and B (where 0° < A + B ≤ 90°).

Section E: Short Answer Questions (3 Marks Each)

21. Prove that: √[(1 + sin θ) / (1 - sin θ)] = sec θ + tan θ.

22. If A = 60° and B = 30°, verify that sin(A - B) = sin A cos B - cos A sin B.

23. Prove that: (sin θ - 2 sin³ θ) / (2 cos³ θ - cos θ) = tan θ.

24. Prove that: 2 (sin⁶ θ + cos⁶ θ) - 3 (sin⁴ θ + cos⁴ θ) + 1 = 0.

25. Given that sin θ + cos θ = x, prove that:
sin⁴ θ + cos⁴ θ = [2 - (x² - 1)²] / 2

Section F: Long Answer Questions (5 Marks Each)

26. If 1 + sin² θ = 3 sin θ cos θ, then prove that tan θ = 1 or 1/2.

OR

Prove that: (sec θ + tan θ - 1) / (tan θ - sec θ + 1) = (1 + sin θ) / cos θ.

27. Prove the identity: [1 / (sec² θ - cos² θ) + 1 / (cosec² θ - sin² θ)] sin² θ cos² θ = (1 - sin² θ cos² θ) / (2 + sin² θ cos² θ).

28. If x = cosec A + cos A and y = cosec A - cos A, prove that [2 / (x + y)]² + [(x - y) / 2]² - 1 = 0.

29. Prove that: (sin A - sin B) / (cos A + cos B) + (cos A - cos B) / (sin A + sin B) = 0.

OR

If tan A + sin A = m and tan A - sin A = n, prove that m² - n² = 4√mn.

30. Using the formula sin A = √[(1 - cos 2A) / 2], find the value of sin 30° given that cos 60° = 1/2.

Section G: Case-Based Questions (4 Marks Each)

Case Study 1: In ΔPQR, ∠Q = 90°, PQ = x, QR = (x + 2), and PR is the hypotenuse. ∠P = θ and ∠R = φ.
(i) Find the value of cos θ. (1 Mark)
(ii) Find (√(x³ + x²)) tan θ. (1 Mark)
(iii) Find (√(x + 1)) cot φ. (2 Marks)

Case Study 2: In the adjoining figure, ∠B = 90°, ∠BAC = θ, BC = CD = 4 cm, and AD = 10 cm.
(i) Find the length of AB. (1 Mark)
(ii) Find the length of AC. (1 Mark)
(iii) Find the values of sin θ and cos θ. (2 Marks)

CBSE Class 10 Mathematics

Chapter: Introduction to Trigonometry