Trigonometry

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

Trigger: the expression contains ${\sin }^2$ and ${\cos }^2$ (or ${\sin }^4$, ${\cos }^4$), or a single squared ratio plus its linear term.

Instances:
(i) substitute to get a quadratic in $\cos$, then factorise (e.g. $(u-1)(u-2)=0$);
(ii) substitute to get ${\sin }^2=$ constant and read off the base angle;
(iii) treat a degree-4 equation as a quadratic in ${\cos }^2$ and factorise;
(iv) clear ${\sin }^2$ from a rational function so the numerator and denominator share a factor that cancels.