Question

How do I simplify (sin^4x-2sin^2x+1)cosx?

`(sin^4X-2sin^2X+1)cosx` I just cant see how to do this. Please give an easy explanation. I was doing so well in math until these types of problems come up and now I am stuck.

Answer 1

`cos^5x`

Explanation:

This type of problem is truly not that bad once you recognize that it involves a little algebra!

First, I'll rewrite the given expression to make the following steps easier to understand. We know that `sin^2x` is just a simpler way to write `(sin x)^2`. Similarly, `sin^4x = (sin x)^4`.

We can now rewrite the original expression.

`(sin^4 x - 2 sin^2 x +1) cos x`

`=[ (sin x)^4 - 2 (sin x)^2 + 1] cos x`

Now, here's the part involving algebra. Let `sin x = a`. We can write `(sin x)^4 - 2 (sin x)^2 + 1` as

`a^4 - 2 a^2 + 1`

Does this look familiar? We just need to factor this! This is a perfect square trinomial. Since `a^2 - 2ab + b^2 = (a-b)^2`, we can say

`a^4 - 2 a^2 + 1 = (a^2 - 1)^2`

Now, switch back to the original situation. Re-substitute `sin x` for `a`.

`[ (sin x)^4 - 2 (sin x)^2 + 1] cos x`

`= [(sin x)^2 -1]^2 cos x`

`= (color(blue)(sin^2x - 1))^2 cos x`

We can now use a trigonometric identity to simplify the terms in blue. Rearranging the identity `sin^2 x + cos^2 x = 1`, we get `color(blue)(sin^2 x -1 = -cos^2x)`.

`=(color(blue)(-cos^2x))^2 cos x`

Once we square this, the negative signs multiply to become positive.

`= (cos^4x) cos x`

`=cos^5x`

Thus, `(sin^4 x - 2 sin^2 x +1) cos x = cos^5x`.