Question

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

Answer 1

To prove : $\sin (A+B)+\sin (A-B)=2\sin A\sin B$

Use formula:

$$\begin{gathered} \sin (A-B)=\sin A.\cos B-\sin B.\cos A \\ \sin (A+B)=\sin A.\cos B+\sin B.\cos A \end{gathered}$$

Solve the L.H.S part.

$$\begin{gathered} \sin (A+B)+\sin (A-B)=\sin A\cos B+\sin B\cos A+\sin A\cos B-\sin B\cos A \\ \Rightarrow =\sin A\cos B+\sin A\cos B \\ \Rightarrow =2\sin A\cos B \end{gathered}$$

Therefore, L.H.S= R.H.S.

Hence, proved.