Work on One Side

To prove an identity, start from one side, usually the more complicated one, and rewrite it until it reads the other. Every step must hold at every angle where the expression is defined: an identity, an expansion, a cancellation. Never move terms across the equals sign as if solving an equation; that assumes the two sides are already equal, which is the thing to prove.

Prove $\csc\left(x\right)-\sin\left(x\right) = \cos\left(x\right)\cot\left(x\right)$. Start on the left. Write csc x as $\frac{1}{\sin\left(x\right)}$, then Write as one fraction over sin x: $\frac{1-{\sin\left(x\right)}^{2}}{\sin\left(x\right)}$. The top is ${\cos\left(x\right)}^{2}$, so the fraction is $\frac{{\cos\left(x\right)}^{2}}{\sin\left(x\right)}$. Split it as $\cos\left(x\right)\cdot \frac{\cos\left(x\right)}{\sin\left(x\right)}$, which is the right side, $\cos\left(x\right)\cot\left(x\right)$.