Selina Solutions Concise Maths Class 10 Chapter 21 Trigonometrical Identities Exercise 21(A)

Understanding different trigonometric ratios and its relations between them to prove various trigonometric identities is the major focus in this exercise. Students who want to learn the right procedures to prove such identities can refer to the Selina Solutions for Class 10 Maths. The solutions are created by subject matter experts at BYJU’S. The solution PDF of the Concise Selina Solutions for Class 10 Maths Chapter 21 Trigonometrical Identities Exercise 21(A) is available in the link provided below.

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Access Selina Solutions Concise Maths Class 10 Chapter 21 Trigonometrical Identities Exercise 21(A)

Prove the following identities:

1. sec A – 1/ sec A + 1 = 1 – cos A/ 1 + cos A

Solution:

– Hence Proved

2. 1 + sin A/ 1 – sin A = cosec A + 1/ cosec A – 1

Solution:

– Hence Proved

3. 1/ tan A + cot A = cos A sin A

Solution:

Taking L.H.S,

– Hence Proved

4. tan A – cot A = 1 – 2 cos² A/ sin A cos A

Solution:

Taking LHS,

– Hence Proved

5. sin⁴ A – cos⁴ A = 2 sin² A – 1

Solution:

Taking L.H.S,

sin⁴ A – cos⁴ A

= (sin² A)² – (cos² A)²

= (sin² A + cos² A) (sin² A – cos² A)

= sin²A – cos²A

= sin²A – (1 – sin²A) [Since, cos² A = 1 – sin² A]

= 2sin² A – 1

– Hence Proved

6. (1 – tan A)² + (1 + tan A)² = 2 sec² A

Solution:

Taking L.H.S,

(1 – tan A)² + (1 + tan A)²

= (1 + tan² A + 2 tan A) + (1 + tan² A – 2 tan A)

= 2 (1 + tan² A)

= 2 sec² A [Since, 1 + tan² A = sec² A]

– Hence Proved

7. cosec⁴ A – cosec² A = cot⁴ A + cot² A

Solution:

cosec⁴ A – cosec² A

= cosec² A(cosec² A – 1)

= (1 + cot² A) (1 + cot² A – 1)

= (1 + cot² A) cot² A

= cot⁴ A + cot² A = R.H.S

– Hence Proved

8. sec A (1 – sin A) (sec A + tan A) = 1

Solution:

Taking L.H.S,

sec A (1 – sin A) (sec A + tan A)

– Hence Proved

9. cosec A (1 + cos A) (cosec A – cot A) = 1

Solution:

Taking L.H.S,

– Hence Proved

10. sec² A + cosec² A = sec² A . cosec² A

Solution:

Taking L.H.S,

– Hence Proved

11. (1 + tan² A) cot A/ cosec² A = tan A

Solution:

Taking L.H.S,

= RHS

– Hence Proved

12. tan² A – sin² A = tan² A. sin² A

Solution:

Taking L.H.S,

tan² A – sin² A

– Hence Proved

13. cot² A – cos² A = cos²A. cot²A

Solution:

Taking L.H.S,

cot² A – cos² A

– Hence Proved

14. (cosec A + sin A) (cosec A – sin A) = cot² A + cos² A

Solution:

Taking L.H.S,

(cosec A + sin A) (cosec A – sin A)

= cosec² A – sin² A

= (1 + cot² A) – (1 – cos² A)

= cot² A + cos² A = R.H.S

– Hence Proved

15. (sec A – cos A)(sec A + cos A) = sin² A + tan² A

Solution:

Taking L.H.S,

(sec A – cos A)(sec A + cos A)

= (sec² A – cos² A)

= (1 + tan² A) – (1 – sin² A)

= sin² A + tan² A = RHS

– Hence Proved

16. (cos A + sin A)² + (cosA – sin A)² = 2

Solution:

Taking L.H.S,

(cos A + sin A)² + (cosA – sin A)²

= cos2 A + sin2 A + 2cos A sin A + cos2 A – 2cosA.sinA

= 2 (cos² A + sin² A) = 2 = R.H.S

– Hence Proved

17. (cosec A – sin A)(sec A – cos A)(tan A + cot A) = 1

Solution:

Taking LHS,

(cosec A – sin A)(sec A – cos A)(tan A + cot A)

= RHS

– Hence Proved

18. 1/ sec A + tan A = sec A – tan A

Solution:

Taking LHS,

= RHS

– Hence Proved

19. cosec A + cot A = 1/ cosec A – cot A

Solution:

Taking LHS,

cosec A + cot A

= RHS

– Hence Proved

20. sec A – tan A/ sec A + tan A = 1 – 2 secA tanA + 2 tan² A

Solution:

Taking LHS,

= 1 + tan² A + tan² A – 2 sec A tan A

= 1 – 2 sec A tan A + 2 tan² A = RHS

– Hence Proved

21. (sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

Solution:

Taking LHS,

(sin A + cosec A)² + (cos A + sec A)²

= sin² A + cosec² A + 2 sin A cosec A + cos² A + sec² A + 2cos A sec A

= (sin² A + cos² A ) + cosec² A + sec² A + 2 + 2

= 1 + cosec² A + sec² A + 4

= 5 + (1 + cot² A) + (1 + tan² A)

= 7 + tan² A + cot² A = RHS

– Hence Proved

22. sec² A. cosec² A = tan² A + cot² A + 2

Solution:

Taking,

RHS = tan² A + cot² A + 2 = tan² A + cot² A + 2 tan A. cot A

= (tan A + cot A)² = (sin A/cos A + cos A/ sin A)²

= (sin2 A + cos2 A/ sin A.cos A)² = 1/ cos² A. sin² A

= sec² A. cosec² A = LHS

– Hence Proved

23. 1/ 1 + cos A + 1/ 1 – cos A = 2 cosec² A

Solution:

Taking LHS,

= RHS

– Hence Proved

24. 1/ 1 – sin A + 1/ 1 + sin A = 2 sec² A

Solution:

Taking LHS,

= RHS

– Hence Proved