Question

Which expression is equivalent to the following?

Answer 1

Step-by-step explanation:

$\sin^{4}x \: - \cos^{4} x = (\sin^{2}x)^{2} - (\cos^{2}x)^{2}$

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

$= \sin^{4}x - (1 - 2\sin^{2}x + \sin^{4}x)$

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

$= - \cos2x$

Note: I used the identity

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

for the last step.

PS. I love proving trigonometric identities!