Answer
$$(1+\sin x+\cos x)^2=2(1+\sin x)(1+\cos x)$$
The expression is an identity.
Work Step by Step
$$(1+\sin x+\cos x)^2=2(1+\sin x)(1+\cos x)$$
We would examine the left side first.
$$A=(1+\sin x+\cos x)^2$$
$$A=[1+(\sin x+\cos x)]^2$$
$$A=1+2(\sin x+\cos x)+(\sin x+\cos x)^2$$
$$A=1+2(\sin x+\cos x)+(\sin^2 x+2\sin x\cos x+\cos^2 x)$$
$$A=1+2\sin x+2\cos x+\sin^2 x+2\sin x\cos x+\cos^2x$$
$$A=1+2\sin x+2\cos x+2\sin x\cos x+(\sin^2x+\cos^2x)$$
$$A=1+2\sin x+2\cos x+2\sin x\cos x+1$$ (for $\sin^2 x+\cos^2x=1$)
$$A=2\sin x+2\cos x+2\sin x\cos x+2$$
Now from the right side,
$$B=2(1+\sin x)(1+\cos x)$$
$$B=2(1+\cos x+\sin x+\sin x\cos x)$$
$$B=2+2\cos x+2\sin x+2\sin x\cos x$$
Therefore, $A=B$. The expression is an identity..
