# Thread: help me verify this identity

1. ## help me verify this identity

The original problem

sin^3x+cos^3x
---------------- = 1-sinxcosx
sinx+cosx

Now I have this awesome book that tells how to do the problems step by step but it doesn't do a very good job of explaining how it went from step 1 to step 2. For example the book gives me

sinx^3+cosx^3 _ _ (sinx+cosx)(sinx^2-sinxcosx+cosx^2)
--------------- = ------------------------------------ = ... = 1-sinxcosx
sinx+cosx _ _ _ _ _ _ _ _ _ sinx+cosx

SO my question is how do you get (sinx^2-sinxcosx+cosx^2) from factoring (sinx+cosx) out of (sinx^3+cox^3).

Oh and the _ _ _ _ _ _ _ _ _ is where spaces would be if the forum didn't cut out multiple spaces

2. Originally Posted by omgwtfbbq
The original problem

sin^3x+cos^3x
---------------- = 1-sinxcosx
sinx+cosx

Now I have this awesome book that tells how to do the problems step by step but it doesn't do a very good job of explaining how it went from step 1 to step 2. For example the book gives me

sinx^3+cosx^3 _ _ (sinx+cosx)(sinx^2-sinxcosx+cosx^2)
--------------- = ------------------------------------ = ... = 1-sinxcosx
sinx+cosx _ _ _ _ _ _ _ _ _ sinx+cosx

SO my question is how do you get (sinx^2-sinxcosx+cosx^2) from factoring (sinx+cosx) out of (sinx^3+cox^3).

Oh and the _ _ _ _ _ _ _ _ _ is where spaces would be if the forum didn't cut out multiple spaces
$\frac{sin ^{3} x + cos ^{3} x}{sin x + cos x} = 1 - (sinx \ cos x)$

Let $sinx = u$ and $cos = v$

LHS:
$\frac{u^3 + v^3}{u + v} = \frac{(u + v)(u^2 - uv + v^2)}{u + v} = u^2 - uv + v^2$

So we have simplified the left side to that.

$u^2 - uv + v^2 = sin ^{2}x + cos ^{2}x - (sinx \ cos x) = 1 - (sinx \ cosx)$

3. Originally Posted by janvdl
$\frac{sin ^{3} x + cos ^{3} x}{sin x + cos x} = 1 - (sinx \ cos x)$

Let $sinx = u$ and $cos = v$

LHS:
$\frac{u^3 + v^3}{u + v} = \frac{(u + v)(u^2 - uv + v^2)}{u + v} = u^2 - uv + v^2$

So we have simplified the left side to that.

$u^2 - uv + v^2 = sin ^{2}x + cos ^{2}x - (sinx \ cos x) = 1 - (sinx \ cosx)$

Smoooooothe!!