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.

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

 

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Access other exercises of Selina Solutions Concise Maths Class 10 Chapter 21 Trigonometrical Identities

Exercise 21(B) Solutions

Exercise 21(C) Solutions

Exercise 21(D) Solutions

Exercise 21(E) Solutions

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:

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 1

– Hence Proved

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

Solution:

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 2

– Hence Proved

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

Solution:

Taking L.H.S,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 3

– Hence Proved

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

Solution:

Taking LHS,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 4

– 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)

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 5

– Hence Proved

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

Solution:

Taking L.H.S,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 6

– Hence Proved

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

Solution:

Taking L.H.S,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 7

– Hence Proved

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

Solution:

Taking L.H.S,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 8

= RHS

– Hence Proved

12. tan2 A – sin2 A = tan2 A. sin2 A

Solution:

Taking L.H.S,

tan2 A – sin2 A

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 9

– Hence Proved

13. cot2 A – cos2 A = cos2A. cot2A

Solution:

Taking L.H.S,

cot2 A – cos2 A

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 10

– 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)

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 11

= RHS

– Hence Proved

18. 1/ sec A + tan A = sec A – tan A

Solution:

Taking LHS,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 12

= RHS

– Hence Proved

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

Solution:

Taking LHS,

cosec A + cot A

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 13

= RHS

– Hence Proved

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

Solution:

Taking LHS,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 14

= 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,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 15

= RHS

– Hence Proved

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

Solution:

Taking LHS,

Selina Solutions Concise Class 10 Maths Chapter 21 ex. 21(A) - 16

= RHS

– Hence Proved

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