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