Answer
Work on the left side of the identity using $\sin^2{\theta}+\cos^2\theta=1$.
Refer to the step-by-step part below for the complete proof.
Work Step by Step
We have to show that:
$(\sin\theta+\cos\theta)^2+(\sin\theta-\cos\theta)^2=2$
By evaluating the left side we get:
$=(\sin^2\theta+2\sin\theta\cos\theta+\cos^2\theta)+(\sin^2\theta-2\sin\theta\cos\theta+\cos^2\theta)\\
=2\sin^2\theta + 2\cos^2\theta\\
=2(\sin^2\theta+\cos^2\theta)$
Using the fact that $\sin^2\theta+\cos^2\theta=1$, the above expression simplifies to
$=2(1)\\
=2$
Since LHS = RHS, the identity's proof is complete.
