ANSWER
B.
[tex] - \cos^{2} x [/tex]
EXPLANATION
The given expression is
(sin x + 1)(sin x − 1)
Note that:
[tex](x + 1)(x - 1) = {x}^{2} - 1[/tex]
This implies that,
[tex]( \sin \: x + 1)( \sin \: x - 1) = \sin^{2} x - 1[/tex]
We can factor -1 on the right hand side to get,
[tex]( \sin \: x + 1)( \sin \: x - 1) = - (1 - \sin^{2} x )[/tex]
Note that from the Pythagorean Identity
[tex]1 - \sin^{2} x = \cos^{2} x[/tex]
We apply this identity to obtain:
[tex]( \sin \: x + 1)( \sin \: x - 1) = - \cos^{2} x [/tex]
The correct choice is B
