Trigonometric identities

The reciprocal, ratio, Pythagorean and complementary-angle identities, and how to simplify trig expressions and prove identities.

The core identities

Ratio identities

$$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}$$

Pythagorean identities

$$\sin^2\theta + \cos^2\theta = 1 \qquad 1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \operatorname{cosec}^2\theta$$

Complementary angles

$$\sin(90^\circ - \theta) = \cos\theta \qquad \cos(90^\circ - \theta) = \sin\theta \qquad \tan(90^\circ - \theta) = \cot\theta$$

sin²θ + cos²θ = 1

Orange: sin²θ. Blue: cos²θ. Pink dashed: their sum, which is 1 for every angle. Hover to check any angle.

Example 1●●

Simplify $\dfrac{1 - \cos^2\theta}{\sin\theta\cos\theta}$.

$= \dfrac{\sin^2\theta}{\sin\theta\cos\theta}$

1 − cos²θ = sin²θ.

$= \dfrac{\sin\theta}{\cos\theta} = \tan\theta$

Example 2●●●

Prove that $(1 + \tan^2\theta)\cos^2\theta = 1$.

LHS $= \sec^2\theta \cos^2\theta$

Pythagorean identity.

$= \dfrac{1}{\cos^2\theta}\cos^2\theta = 1 =$ RHS

sec θ = 1/cos θ.

Flashcards

sin²θ + cos²θ

$1$

1 + tan²θ

$\sec^2\theta$

1 + cot²θ

$\operatorname{cosec}^2\theta$

cos(90° − θ)

$\sin\theta$

Proving an identity

Work on one side until it matches the other.

Practice

$1 - \sin^2\theta =$
$\sec^2\theta - \tan^2\theta =$
$\sin 25^\circ$ equals:
$\cos\theta\tan\theta =$
Evaluate $\sin^2 37^\circ + \cos^2 37^\circ$ without a calculator.
Evaluate $\cos^2 20^\circ + \cos^2 70^\circ$ without a calculator.
$\dfrac{\sin\theta}{\tan\theta} =$
Which is the correct way to start proving $\dfrac{1}{\cos^2\theta} - 1 = \tan^2\theta$?