Question

How do you simplify the expression `sin^4theta-cos^4theta`?

Answer 1

`sin^4theta-cos^4theta=-cos2theta`

Explanation:

`sin^4theta-cos^4theta`

= `(sin^2theta)^2-(cos^2theta)^2`
(and using `(a^2-b^2)=(a+b)(a-b)`, this becomes)

= `(sin^2theta-cos^2theta)(sin^2theta+cos^2theta)`

= `(sin^2theta-cos^2theta)*1`

= `-(cos^2theta-sin^2theta)`
(as `cos^2A-sin^2A=cos2A)`

= `-cos2theta`