Question

How do I prove that `(3-6 cos^2 x)/(sin x- cos x)` = 3(sin x + cos x)?

Answer 1

Yes, the equality is correct. See explanation.

Explanation:

You have
`frac(3-6cos^2 x)(sin x - cos x)`

On the numerator, factor out 3:
`= 3*frac(1-2cos^2 x)(sin x - cos x)`

Remember the identity `sin^2 x + cos^2 x = 1`.
Substitute the 1 in the numerator:
`= 3*frac(sin^2 x + cos^2 x -2cos^2 x)(sin x - cos x)`

Simplify by adding the cos^2 x together:
`= 3*frac(sin^2 x -cos^2 x)(sin x - cos x)`

Remember that when you have something of the form `a^2 - b^2 = (a-b)(a+b)`

So rewrite the numerator
`= 3*frac((sin x -cos x)(sin x + cos x))(sin x - cos x)`

And remove `sin x - cos x` from the numerator and denominator. You are left with:

`=3(sin x + cos x)`

Q.E.D.