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# Class 11 RD Sharma Solutions – Chapter 8 Transformation Formulae – Exercise 8.2 | Set 2

• Last Updated : 08 May, 2021

### Question 11. If cosec A + sec A = cosec B + sec B, prove that tan A tan B =.

Solution:

We have, cosec A + sec A = cosec B + sec B

=> sec A âˆ’ sec B = cosec B âˆ’ cosec A

=>

=>

=>

=> tan A tan B =

=> tan A tan B =

Hence proved.

### Question 12. If sin 2A = Î» sin 2B, prove that.

Solution:

We are given, sin 2A = Î» sin 2B

=>

On applying Componendo and Dividendo, we get,

=>

=>

=>

=>

=>

Hence proved.

Solution:

We have,

L.H.S. =

=

=

=

=

=

=

=

= cot C

= R.H.S.

Hence proved.

### (ii) sin (Bâˆ’C) cos (Aâˆ’D) + sin (Câˆ’A) cos (Bâˆ’D) + sin (Aâˆ’B) cos (Câˆ’D) = 0

Solution:

We have, L.H.S. = sin (Bâˆ’C) cos (Aâˆ’D) + sin (Câˆ’A) cos (Bâˆ’D) + sin (Aâˆ’B) cos (Câˆ’D)

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 14. If, prove that tan A tan B tan C tan D = âˆ’1.

Solution:

We have,

=>

=>

=>

=>

=>

=>

=>

=>. . . . (1)

Also,

=>

=>

=>

=>

=>. . . . (2)

Dividing (1) by (2), we get,

=>

=>

=>

=> tan A tan B tan C tan D = âˆ’1

Hence proved.

### Question 15. If cos (Î±+Î²) sin(Î³+Î´) = cos (Î±âˆ’Î²) sin(Î³âˆ’Î´), prove that cot Î± cot Î² cot Î³ = cot Î´.

Solution:

We have, cos (Î±+Î²) sin(Î³+Î´) = cos (Î±âˆ’Î²) sin(Î³âˆ’Î´)

=>

=>

=>

=>

=>. . . . (1)

Also,

=>

=>

=>

=>. . . . (2)

Dividing (1) by (2), we get,

=>

=>

=> cot Î± cot Î² = tan Î³ cot Î´

=> cot Î± cot Î² cot Î³ = cot Î´

Hence proved.

### Solution:

Given, y sin Ã˜ = x sin (2Î¸ + Ã˜)

=>

On applying Componendo and Dividendo, we get,

=>

=>

=>

=>

=> tan (Ã˜+Î¸) cot Î¸ =

=>

=> (y âˆ’ x) cot Î¸ = (x + y) cot (Î¸ + Ã˜)

=> (x + y) cot (Î¸ + Ã˜) = (y âˆ’ x) cot Î¸

Hence proved.

### Question 17. If cos (A+B) sin (Câˆ’D) = cos (Aâˆ’B) sin (C+D), prove that tan A tan B tan C + tan D = 0.

Solution:

We are given, cos (A+B) sin (Câˆ’D) = cos (Aâˆ’B) sin (C+D)

=>

On applying Componendo and Dividendo, we get,

=>

=>

=>

=>

=> âˆ’tan D = tan A tan B tan C

=> tan A tan B tan C + tan D = 0

Hence proved.

### Question 18. If , prove that xy + yz + zx = 0.

Solution:

We have,= k (say)

x =

y =

z =

So, L.H.S. = xy + yz + zx

=

=

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 19. If m sin Î¸ = n sin (Î¸ + 2a), prove that.

Solution:

We are given, m sin Î¸ = n sin (Î¸ + 2a)

=>

On applying Componendo and Dividendo, we get,

=>

=>

=>

=>

=>

Hence, proved.

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