Trigonometry

Before solving any trig equation, reduce it to one of the three canonical shapes and write every solution at once: $\sin A=\sin B\Rightarrow A=B+2k\pi$ or $A=\pi -B+2k\pi$; $\cos A=\cos B\Rightarrow A=\pm B+2k\pi$; $\tan A=\tan B\Rightarrow A=B+k\pi$, then keep only the roots in range.

Trigger: any equation reducible to equality of two like trig ratios, especially when a full family or a sketch of all solutions is wanted.

Instances:
(i) $\sin y=\sin x$ gives the lines $y=x+2k\pi$ and $y=\pi -x+2k\pi$.
(ii) $\cos (x-\alpha )=\cos \beta$ gives $x-\alpha =\pm \beta +2k\pi$, hence $\tan x=\tan (\alpha \pm \beta )$.
(iii) A $\tan$ equation needs the $+k\pi$ period, not $2k\pi$, or solutions are lost.