Question

Simplify $\sqrt{1+\sin 2x}$

Answer 1

$\sqrt{1+\sin 2x}$

$\begin{array}{l}=\sqrt{{\sin }^{2}x+{\cos }^{2}x+2\sin x\times \cos x}\end{array}$ $\left[\because {\sin }^{2}x+{\cos }^{2}x=1,\sin 2x=2\sin x\cos x\right]$

$\begin{array}{l}=\sqrt{(\sin x+\cos x{)}^2}\end{array}$

$\begin{array}{l}=\sin x+\cos x\end{array}$

Therefore, $\sqrt{1+\sin 2x}$$\begin{array}{l}=\sin x+\cos x\end{array}$