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

1 Answer
Sep 20, 2015

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.