Proving trigonometric identities

Work on one side at a time, usually the more complicated one, until it becomes the other side. Never move terms across the equals sign, because that assumes what you are proving.

Useful moves:

• Write everything in terms of $\sin$ and $\cos$, using $\tan x=\frac{\sin x}{\cos x}$.
• Replace $1$ with ${\sin }^{2}x+{\cos }^{2}x$ where it helps.
• Expand double and compound angles.
• Find a common denominator, then factorise and cancel.

Example. Prove $\frac{1-{\cos }^{2}x}{\sin x\cos x}=\tan x$.

$\text{LHS}=\frac{{\sin }^{2}x}{\sin x\cos x}=\frac{\sin x}{\cos x}=\tan x=\text{RHS}$

If you are stuck on the LHS, simplify the RHS as well and meet in the middle. Show both sides clearly.

An identity is undefined wherever a denominator is 0 or $\tan$ is undefined.

Example. $\frac{1-{\cos }^{2}x}{\sin x\cos x}=\tan x$ is undefined when:

• $\sin x=0$: $x=k\cdot 180°$
• $\cos x=0$: $x=90°+k\cdot 180°$

Together: $x=k\cdot 90°$, $k\in \mathbb{Z}$.

Check every denominator in the original identity, including those on the right-hand side. A common error is checking only the simplified version, where the problem factor has already cancelled.