Rewrite in sin and cos

When no identity fits the expression as it stands, write every function in sines and cosines. It is rarely the shortest route, but it always gives something to work with.

Prove ${\cot\left(x\right)}^{2}{\sin\left(x\right)}^{2}+{\sin\left(x\right)}^{2} = 1$. Write cot x as $\frac{\cos\left(x\right)}{\sin\left(x\right)}$, so the first term is ${\frac{\cos\left(x\right)}{\sin\left(x\right)}}^{2}\cdot {\sin\left(x\right)}^{2}$. Square the fraction, $\frac{{\cos\left(x\right)}^{2}}{{\sin\left(x\right)}^{2}}$, multiply the tops and the bottoms, and cancel ${\sin\left(x\right)}^{2}$: the first term is now ${\cos\left(x\right)}^{2}$. The line reads ${\cos\left(x\right)}^{2}+{\sin\left(x\right)}^{2}$, which is 1.

Squaring a quotient squares its top and its bottom: ${\frac{a}{b}}^{2} = \frac{{a}^{2}}{{b}^{2}}$. Forgetting to square the bottom is the slip to watch for.