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