Trigonometry, rewriting an expression

Question

How do I rewrite $((\cos t)^3) - 2((\cos t)\cdot((\sin t)^2))$ to $3(\cos t)^3 - 2 \cos t$?

Answer 1

First, we denote $(\cos t)^3$ as $\cos^3 t$, and $(\sin t)^2 = \sin^2 t$.

Then we use the identity $$\sin^2 t + \cos^2 t = 1 \implies \sin^2 t = 1-\cos^2 t$$ and simplify.

$$\cos^3 t- 2\cos t\sin^2 t = \cos^3 t - 2\cos t(1-\cos^2 t) $$ $$=\cos^3 t - 2\cos t + 2\cos^3 t$$ $$= 3\cos^3 t - 2\cos t$$