Question

Prove that Sin (A+B) -- 2SinA + Sin (A+B) / Cos (A+B) -- 2Cos A + Cos( A--B)= Tan A

Answer 1

Recall the identities,

$\sin (A+B)+\sin (A-B)=2\sin A \cos B\\ \cos (A+B)+\sin (A-B)=2\cos A \cos B$

Here,

$LHS=\frac{\sin (A+B)-2\sin A+\sin (A-B)}{\cos (A+B)-2\cos A+\sin (A-B)}\\ LHS=\frac{2\sin A \cos B-2\sin A}{2\cos A \cos B-2\cos A }\\ LHS=\frac{2\sin A( \cos B-1)}{2\cos A( \cos B-1) }\\ LHS=\frac{\sin A}{\cos A }\\ LHS=\tan A=RHS$