Question

verify the identity

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

Answer 1

Answer:

See below for proof of identity.

Step-by-step 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 $b=\cos\theta$:

$\displaystyle \frac{\sin^3\theta+\cos^3\theta}{\sin\theta+\cos\theta}\\\\\frac{(\sin\theta+\cos\theta)(\sin^2\theta-\sin\theta\cos\theta+\cos^2\theta)}{\sin\theta+\cos\theta}\\ \\\frac{(\sin\theta+\cos\theta)(1-\sin\theta\cos\theta)}{\sin\theta+\cos\theta}\\\\1-\sin\theta\cos\theta$

Thus, the identity is proven.