Trigonometry

When an equation mixes even powers of $\sin$ and $\cos$, replace ${\sin }^2$ by $1-{\cos }^2$ (or vice versa) so that only ONE trig ratio remains, then treat that ratio as an algebraic unknown and factorise the polynomial.

Trigger: the expression contains mixed squares like ${\sin }^2$ and ${\cos }^2$, or a single squared ratio plus its linear term.

Instances:
(i) substitute $\tan x=\frac{\sin x}{\cos x}$ and multiply out to expose the ${\sin }^{2}x$ and ${\cos }^{2}x$ terms;
(ii) substitute to get a quadratic in $\cos x$, then factorise;
(iii) apply it inside a bracket after factoring out a common term.