Question

How do you simplify `\frac{\sin^4 \theta - \cos^4 \theta}{\sin^2 \theta - \cos^2 \theta}` using the trigonometric identities?

Answer 1

You should get: `(sin^4(theta)-cos^4(theta))/(sin^2(theta)-cos^2(theta))=1`

This is because in the numerator you have:

`sin^4(theta)-cos^4(theta)` which is the same as:

`[sin^2(theta)+cos^2(theta)]*[sin^2(theta)-cos^2(theta)]`

(Remember that `(a+b)*(a-b)=a^2-b^2`)

So your fraction becomes:

`([sin^2(theta)+cos^2(theta)]*[sin^2(theta)-cos^2(theta)])/(sin^2(theta)-cos^2(theta))=`

You can now simplyfy the two: `sin^2(theta)-cos^2(theta)` in the numerator and denominator.
You are left with:
`[sin^2(theta)+cos^2(theta)]` which is always equal to 1 (for every angle `theta`)