Question

What is $\sin \left(x\right)\cos \left(x\right)$?

Answer 1

Method $1$:

We know that,

$2\sin x\cos x=\sin 2x$

Divide both sides by $2$, we get

$\sin x\cos x=\frac{1}{2}\left(\sin 2x\right)$

Method $2$:

We know that,

$\sin \left(A+B\right)=\sin A\cos B+\cos A\sin B$

substitute $A=B=x$, we get

$\sin \left(x+x\right)=\sin x\cos x+\cos x\sin x$

$\Rightarrow$ $\sin 2x=2\sin x\cos x$

$\Rightarrow$ $\frac{1}{2}\left(\sin 2x\right)=\sin x\cos x$