Trigonometric Identities Survival Kit

There are dozens of trig identities, but in practice you only need to memorise about a dozen — the rest can be derived in seconds from those. This page is the survival kit: every identity that earns its keep, with short worked examples for each.

The Pythagorean trio

$\sin^2\theta + \cos^2\theta = 1$
$1 + \tan^2\theta = \sec^2\theta$
$1 + \cot^2\theta = \csc^2\theta$

The first is the most-used identity in all of mathematics. The other two are obtained by dividing through by $\cos^2$ or $\sin^2$.

Sum and difference formulas

$\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta$
$\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta$

Mnemonic for cos: "cos cos minus sin sin" with opposite sign — sin is "sin cos plus cos sin" with same sign.

Double angle formulas

Substitute $\alpha = \beta = \theta$ into the sum formulas:

$\sin(2\theta) = 2\sin\theta \cos\theta$
$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$
$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Three forms of the cosine version exist because of the Pythagorean identity. Pick whichever matches the rest of your expression.

Half angle formulas

Solving the cosine double-angle for $\sin^2$ and $\cos^2$ gives:

$\sin^2\theta = \frac{1 - \cos(2\theta)}{2}, \quad \cos^2\theta = \frac{1 + \cos(2\theta)}{2}$

These are the power-reduction identities — they are how $\int \sin^2 x \, dx$ becomes elementary.

Worked example: simplification

Simplify $\frac{\sin(2x)}{1 + \cos(2x)}$.

1. Numerator: $\sin(2x) = 2\sin x \cos x$.
2. Denominator: $1 + \cos(2x) = 1 + (2\cos^2 x - 1) = 2\cos^2 x$.
3. Quotient: $\frac{2\sin x \cos x}{2\cos^2 x} = \frac{\sin x}{\cos x} = \tan x$.

The whole hairy expression collapses to $\tan x$.

Common mistakes

• Sign errors in sum formulas — write the formula out, don't trust memory mid-problem.
• $\sin^2\theta$ means $(\sin\theta)^2$, not $\sin(\sin\theta)$.
• Forgetting that $2\theta$ is the angle, not 2 times the value — $\sin(2 \cdot 30°) = \sin 60°$, not $2\sin 30°$.

Try with the AI Trigonometry Solver

The Trigonometry Solver takes any expression and applies all of these identities to simplify or solve it.

Related references:

• Simplify Calculator — same simplification ideas, polynomial flavour
• Integral Calculator — power reduction is critical for trig integrals
• Series Calculator — Taylor expansions of sin and cos use these directly