Class 11th Maths – RD Sharma Chapter 5 : trigonometric Functions Exercise 5.1 Solutions

1. Prove the following identity : sec⁴x – sec²x = tan⁴x + tan²x Watch Solution
2. Prove the following identity : sin⁶x + cos⁶x = 1 – 3 sin²x cos²x  Watch Solution
3. Prove the following identity : (cosec x – sin x) (sec x – cos x) (tan x + cot x) = 1 Watch Solution
4. Prove the following identity : cosec x (sec x – 1) – cot x(1 – cos x) = tan x – sin x Watch Solution
5. Prove the following identity : [(1 – sin x cos x) / {cos x (sec x – cosec x)}] · [(sin²x – cos²x) / (sin³x + cos³x)] = sin x Watch Solution
6. Prove the following identity : tan x/(1 – cot x) + cot x/(1 – tan x) = (sec x cosec x + 1) Watch Solution
7. Prove the following identity : (sin³x + cos³x) / (sin x + cos x) + (sin³x – cos³x) / (sin x – cos x) = 2 Watch Solution
8. Prove the following identity : (sec x sec y + tan x tan y)² – (sec x tan y + tan x sec y)² = 1 Watch Solution
9. Prove the following identity : cos x/(1 – sin x) = (1 + cos x + sin x)/(1 + cos x – sin x) Watch Solution
10. Prove the following identity : tan³x / (1 + tan²x) + cot³x / (1 + cot²x) = (1 – 2 sin²x cos²x) / (sin x cos x) Watch Solution
11. Prove the following identity : 1 – [sin²x / (1 + cot x)] – [cos²x / (1 + tan x)] = sin x cos x Watch Solution
12. Prove the following identity : [1 / (sec²x – cos²x) + 1 / (cosec²x – sin²x)] sin²x cos²x = (1 – sin²x cos²x) / (2 + sin²x cos²x) Watch Solution
13. Prove the following identity : (1 + tan α tan β)² + (tan α – tan β)² = sec²α sec²β Watch Solution
14. Prove the following identity : [(1 + cot x + tan x)(sin x – cos x)] / (sec³x – cosec³x) = sin²x cos²x Watch Solution
15. Prove the following identity : (2 sin x cos x – cos x) / (1 – sin x + sin²x – cos²x) = cot x Watch Solution
16. Prove the following identity : cos x (tan x + 2) (2 tan x + 1) = 2 sec x + 5 sin x Watch Solution
17. If a = 2 sin x/(1+ cos x + sin x), then prove that (1 – cos x + sin x)/(1 + sin x) is also equal to a. Watch Solution
18. If sin x = (a² – b²) / (a² + b²), find the values of tan x, sec x and cosec x Watch Solution
19. If tan x = b/a, then find the value of √(a+b)/(a-b) + √(a-b)/(a+b). Watch Solution
20. If tan x = a/b, show that (a sin x – b cos x) / (a sin x + b cos x) = (a² – b²) / (a² + b²) Watch Solution
21. If cosec x – sin x = a³, sec x – cos x = b³, then prove that a²b²(a² + b²) = 1 Watch Solution
22. If cot x (1 + sin x) = 4m and cot x (1 – sin x) = 4n, prove that (m² – n²)² = mn. Watch Solution
23. If sin x + cos x = m, then prove that sin⁶x + cos⁶x = [4 – 3(m² – 1)²] / 4, where m² ≤ 2 Watch Solution
24. If a = sec x – tan x and b = cosec x + cot x, then show that ab + a – b + 1 = 0 Watch Solution
25. Prove that : |√{(1-sin x)/(1+sin x)} + √{(1+sin x)/(1-sin x)}| = -2/cos x, where π/2 < x < π Watch Solution
26. If Tn = sin^n x + cos^n x, prove that (T3 – T5)/T1 = (T5 – T7)/T3 Watch Solution
27. If Tn = sin^n x + cos^n x, prove that 2T6 – 3T4 + 1 = 0 Watch Solution
28. If Tn = sin^n x + cos^n x, prove that 6T10 – 15T8 + 10T8 + 10T6 – 1 = 0 Watch Solution