Question

Verify csc^4x-cot^4x=2csc^2x-1

Answer 1

First, factor the left side:

$(csc^2(x))^2-(cot^2(x))^2$

This can be simplified using $x^2-y^2 = (x+y)(x-y)$ where the csc term is x and the cot term is y:

$(csc^2(x)+cot^2(x))(csc^2(x)-cot^2(x))$

One of our Pythagorean Identities states that:

$csc^2(x)-cot^2(x)=1$

So the left side is now:

$csc^2(x)+cot^2(x)$

Now we can use the same Pythagorean Identity to change the right side to:

$2csc^2(x)-(csc^2(x)-cot^2(x))$

All we did was express 1 in terms of csc and cot.

Distribute the negative:

$2csc^2(x)-csc^2(x)+cot^2(x)$

Combine like terms:

$csc^2(x)+cot^2(x)$

Now the left and right sides are the same.