Question

Which expression is equivalent to (sin x + 1)(sin x − 1)? A. cos2x B. -cos2x C. cos2x + 1 D. cos2x − 1 E. -cos2x + 1

Answer 1

ANSWER

B.

$- \cos^{2} x$

EXPLANATION

The given expression is

(sin x + 1)(sin x − 1)

Note that:

$(x + 1)(x - 1) = {x}^{2} - 1$

This implies that,

$( \sin \: x + 1)( \sin \: x - 1) = \sin^{2} x - 1$

We can factor -1 on the right hand side to get,

$( \sin \: x + 1)( \sin \: x - 1) = - (1 - \sin^{2} x )$

Note that from the Pythagorean Identity

$1 - \sin^{2} x = \cos^{2} x$

We apply this identity to obtain:

$( \sin \: x + 1)( \sin \: x - 1) = - \cos^{2} x$

The correct choice is B