Question

Verify the identity

sin^3 0 +cos^3 0 / sin 0+ cos 0 = 1 - sin 0 cos 0

Answer 1

See below for proof of identity.

Explanation:

Since both terms of
`\sin^3\theta+\cos^3\theta` are perfect cubes, we can use the formula
`a^3+b^3=(a+b)(a^2-ab+b^2)` where
`a=\sin\theta` and
`b=\cos\theta`:

`\displaystyle (\sin^3\theta+\cos^3\theta)/(\sin\theta+\cos\theta)\\\\((\sin\theta+\cos\theta)(\sin^2\theta-\sin\theta\cos\theta+\cos^2\theta))/(\sin\theta+\cos\theta)\\ \\((\sin\theta+\cos\theta)(1-\sin\theta\cos\theta))/(\sin\theta+\cos\theta)\\\\1-\sin\theta\cos\theta`

Thus, the identity is proven.