Answer
$$(\sin x+\cos x)^2=\sin2x+1$$
The equation has been verified to be an identity as below.
Work Step by Step
$$(\sin x+\cos x)^2=\sin2x+1$$
We take from the left side first.
$$X=(\sin x+\cos x)^2$$
Recall the formula $(a+b)^2=a^2+2ab+b^2$ and apply here with $a=\sin x$ and $b=\cos x$
$$X=\sin^2x+2\sin x\cos x+\cos^2x$$
$$X=(\sin^2x+\cos^2x)+(2\sin x\cos x)$$
- Pythagorean Identity: $\sin^2x+\cos^2x=1$
- Double-Angle Identity: $2\sin x\cos x=\sin2x$
Therefore, $$X=1+\sin 2x$$
Thus the left and right side are equal to each other. The equation is an identity.
