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Question

Prove that:
(i) sin A+sin Bsin A-sin B = tan A+B2 cot A -B2

(ii) sin A+sin 3Acos A-cos 3A=cot A

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

(iv) cos A+cos Bcos B-cos A=cot A+B2 cot A-B2

(v) sin A-sin Bcos A+cos B=tanA-B2

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Solution

(i)

Consider LHS: sin A + sin Bsin A - sin B= 2sin A + B2 cos A - B22sin A + B2 cos A - B2 sin A + sin B = 2sin A + B2 cos A - B2, and sin A - sin B = 2sin A - B2 cosA + B2= sin A + B2 cos A - B2sin A - B2 cos A + B2= tan A + B2 cot A - B2=RHSHence, LHS=RHS.

(ii)
Consider LHS: sin A + sin 3Acos A - cos 3A= 2sin A + 3A2 cos A - 3A22sin A + 3A2 sin 3A - A2 sin A + sin B = 2sin A + B2 cos A - B2, and cos A - cos B = 2sin A + B2 cos B - A2=sin 2A cos -Asin 2A sin A= sin 2A cos Asin 2A sin A= cot A=RHSHence, LHS = RHS.

(iii)

Consider LHS:sin 9A - sin 7Acos 7A - cos 9A= 2sin 9A - 7A2 cos 9A + 7A22sin 7A + 9A2 sin 9A - 7A2 sin A - sin B = 2sin A - B2 cos A + B2 and cos A - cos B = 2sin A + B2 cos B - A2=sin A cos 8Asin 8A sin A=cot 8A=RHSHence, LHS = RHS.

(iv)

Consider LHS: cos A + cos Bcos B - cos A= 2cos A - B2 cos A + B22sin A + B2 sin A - B2 cos A + cos B = 2cos A - B2 cos A + B2 and cos A - cos B = 2sin A + B2 cos B - A2= cos A - B2 cos A + B2sin A + B2 sin A - B2= cot A + B2 cot A - B2=RHSHence, LHS=RHS.

(v)
Consider LHS: sin A - sin Bcos A + cos B= 2sin A - B2 cos A + B22cos A + B2 cos A - B2 sin A - sin B = 2sin A - B2 cos A + B2 and cos A + cos B = 2cos A + B2 cos A - B2= sin A - B2 cos A + B2cos A + B2 cos A - B2= tanA - B2=RHSHence, LHS = RHS.

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