Answer
Showed that given statement, $ (\sin\theta - \cos\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-
$ (\sin\theta - \cos\theta)^{2} $ - $1$ = - $2\sin\theta\cos\theta$
Left Side = $ (\sin\theta - \cos\theta)^{2} $ - $1$
= $\sin^{2}\theta - 2 \sin\theta \cos\theta + \cos^{2}\theta$ - $1$
[We know that, $ (a- b)^{2} $ = $a^{2} - 2 ab + b^{2}$]
= $\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,$ (\sin\theta - \cos\theta)^{2} $ - $1$ = - $2\sin\theta\cos\theta$,
is an identity.
