sin2u+cos2u=1, sec2u=1+tan2u, cosec2u=1+cot2u. Use a common argument throughout. These identities replace two squared ratios by one algebraic quantity. The latter two come from dividing the first by cos2u or sin2u, wherever the denominator is nonzero. 1. Record restrictions from the original equation. 2. Replace a squared ratio. 3. Factor or solve the resulting quadratic. 4. Find every angle in the supplied interval. Before dividing by sine or cosine, test its zero case in the original equation. Example: sec2x=1+tan x becomes tan2x-tan x=0. On 0 <= x<360, x=0,45,180,225. Exam check: dividing the example by tan x loses the zero branch. When cos2x=3sin2x, check cos x=0 first, then divide safely to obtain tan2x=1/3.

\(\sin^2u+\cos^2u=1\), \(\sec^2u=1+\tan^2u\), \(\operatorname{cosec}^2u=1+\cot^2u\). Use a common argument throughout.

These identities replace two squared ratios by one algebraic quantity. The latter two come from dividing the first by \(\cos^2u\) or \(\sin^2u\), wherever the denominator is nonzero.

1. Record restrictions from the original equation. 2. Replace a squared ratio. 3. Factor or solve the resulting quadratic. 4. Find every angle in the supplied interval. Before dividing by sine or cosine, test its zero case in the original equation.

Example: \(\sec^2x=1+\tan x\) becomes \(\tan^2x-\tan x=0\). On \(0^\circ\le x<360^\circ\), \(x=0^\circ,45^\circ,180^\circ,225^\circ\).

Exam check: dividing the example by \(\tan x\) loses the zero branch. When \(\cos^2x=3\sin^2x\), check \(\cos x=0\) first, then divide safely to obtain \(\tan^2x=\frac{1}{3}\).

Reducing mixed trigonometric equations

Reducing mixed trigonometric equations