Question

Prove that:

(i) (sin A + sin 3A)/(cos A - cos 3A) = cot A

(iii) (sin 9A - sin 7A)/(cos 7A - cos 9A) = cot 8A

(iii) (sin A - sin B)/(cos A + cos B) = tan(A - B)/2

Answer 1

(i) Let us consider the LHS

(sin A + sin 3A)/(cos A - cos 3A)

On using the formulas,

sin A + sin B = 2 sin(A + B)/2 cos(A - B)/2

cos A - cos B = - 2 sin(A + B)/2 sin(A - B)/2

Therefore now,

= cot A

= RHS

Thus proved.

(ii) Let us consider the LHS

(sin 9A - sin 7A)/(cos 7A - cos 9A)

On using the formulas

sin A + sin B = 2 sin(A + B)/2 cos(A - B)/2

cos A - cos B = - 2 sin(A + B)/2 sin(A - B)/2

Therefore now,

= cot A

= RHS

Thus proved.

(iii) Let us consider the LHS

(sin A - sin B)/(cos A + cos B)

On using the formulas,

sin A - sin B = 2 sin(A + B)/2 cos(A - B)/2

cos A + cos B = - 2 sin(A + B)/2 sin(A - B)/2

Therefore now,

= RHS

Thus proved.