Trig Identities?

Question

Establish the identity

csc^4 (x) - csc^2 (x) = cot^4 (x) + cot^2 (x)

ALSO:

I need someone to carefully and easily explain how to break down trig identities with powers greater than 2.

For example, above we csc^4 (x) and cot^4 (x). How do I play with trig identities whose power is greater than 2?

Answer 1

Prove the identity

csc^4 (x) - csc²x = cot^4 (x) + cot²x

Let's start with the right hand side.

Right Hand Side = cot^4 (x) + cot²x

= cot²x (cot²x + 1) = (cos²x/sin²x)(cos²x/sin²x + sin²x/sin²x)

= [(1 - sin²x)/sin²x] [(cos²x + sin²x)/sin²x]

= [(1 - sin²x)/sin²x] [1/sin²x]

= [(1 - sin²x)/sin^4 (x)] = 1/sin^4 (x) - 1/sin²x

= csc^4 (x) - csc²x = Left Hand Side

Answer 2

Hi,

Prove csc^4 (x) - csc² (x) = cot^4 (x) + cot² (x)

When you have large powers of trig functions, look for ways to factor them to get smaller exponents.

On the left side, factor our csc²(x) as the GCF:

csc² (x)[csc² (x) - 1]

Since 1 + cot²(x) = csc² (x), replace csc² (x) with 1 + cot²(x).

{1 + cot²(x)}[1 + cot²(x) - 1] =
{1 + cot²(x)}[cot²(x)] =
cot²(x) + cot^4 (x)

This is the right side, so the identity is true.

I hope that helps!! :-)