Which expression is equivalent to the following? sin^4(x)-cos^4(x) This is basically calculus......

Question

Which expression is equivalent to the following?


`sin^4(x)-cos^4(x)`

This is basically calculus......

Answer 1

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 = (1)/(2) (1 - \cos \: 2x)`

for the last step.

PS. I love proving trigonometric identities!