Answer
Showed that given statement, $ (\cos\theta + \sin\theta)^{2} $ - $1$ = $2\sin\theta\cos\theta$,
is an identity as left side transforms into right side.
Work Step by Step
Given statement is-
$ (\cos\theta + \sin\theta)^{2} $ - $1$ = $2\sin\theta\cos\theta$
Left Side = $ (\cos\theta + \sin\theta)^{2} $ - $1$
= $\sin^{2}\theta + 2 \sin\theta \cos\theta + \cos^{2}\theta$ - $1$
= $\sin^{2}\theta + \cos^{2}\theta + 2 \sin\theta \cos\theta $ - $1$
= $1$ + $2\sin\theta\cos\theta$ - $1$
[ From first Pythagorean identity]
= $2 \sin\theta\cos\theta $
= Right Side
i.e. Left Side transforms into Right Side
i.e. Given statement, $ (\cos\theta + \sin\theta)^{2} $ - $1$ = $2\sin\theta\cos\theta$,
is an identity.
