Prove the trig equation

Question

Prove: (sin^4 x-9)/(cos^2 x+2)=cos^2 x-4

Thanks!

Answer 1

Hi Nick,

This is a matter of factoring.

Remember that a²-b²=(a-b)(a+b)

So the numerator sin⁴x - 9 = (sin²x -3)(sin²x +3)

Each sin²x can be replaced with (1 - cos²x), so the numerator is now

(1 - cos²x - 3)(1 - cos²x + 3) = (-2-cos²x)(4-cos²x)

And finally, the numerator on the left is

-1(2+cos²x)(4-cos²x)

Rewrite the denominator on the left side as (2+cos²x)

and now our left side fraction is

-1(2+cos²x)(4-cos²x)

-----------------------------

(2+cos²x)

Now the "2+cos²x"s cancel, and the left side is now

-1(4-cos²x)

Distibuting the "-1" in, we get the left side to be

cos²x - 4 which is equal to the right side. And you have proved it! :-)