Question

How do I solve this problem? Prove Identities or simplify using sum and difference formula.

Answer 1

One of the trigonometric identities is :

`\sin (A+B)=\sin A\cos B+\cos A\sin B`

From the problem, we have :

`\sin ((3\pi)/(2)+x)`

Applying the identity above,

`\sin ((3\pi)/(2)+x)=\sin (3\pi)/(2)\cos x+\cos (3\pi)/(2)\sin x`

Note that :

`\begin{gathered} \sin (3\pi)/(2)=-1 \\ \text{and} \\ \cos (3\pi)/(2)=0 \end{gathered}`

The identity will be :

`\begin{gathered} \sin (3\pi)/(2)\cos x+\cos (3\pi)/(2)\sin x \\ \Rightarrow(-1)(\cos x)+(0)(\sin x) \\ \Rightarrow-\cos x \end{gathered}`

The answer is -cos x