Trig Identities

Pythagorean identities

Straight from the unit circle, $\sin^2 \theta + \cos^2 \theta = 1$. Dividing through by $\cos^2$ or $\sin^2$ gives two siblings:

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

where $\sec \theta = 1/\cos \theta$, $\csc \theta = 1/\sin \theta$, $\cot \theta = 1/\tan \theta$.

Reciprocal and quotient identities

$\tan \theta = \dfrac{\sin \theta}{\cos \theta} \qquad \cot \theta = \dfrac{\cos \theta}{\sin \theta}$ $\sec \theta = \dfrac{1}{\cos \theta} \qquad \csc \theta = \dfrac{1}{\sin \theta}$

Even and odd

$\sin(-x) = -\sin x \quad (\text{odd})$ $\cos(-x) = \cos x \quad (\text{even})$ $\tan(-x) = -\tan x \quad (\text{odd})$

Sum and difference

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

Memory tip. Sine: same-different (sin-cos and cos-sin); signs match the $\pm$. Cosine: same-same and different-different (cos-cos minus sin-sin); signs flip.

Double-angle

A special case of the sum identities where $A = B$:

$\sin(2\theta) = 2 \sin \theta \cos \theta$ $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 1 - 2 \sin^2 \theta = 2\cos^2 \theta - 1$

The three forms of $\cos(2\theta)$ are all equivalent via the Pythagorean identity — use whichever is most convenient.

Worked simplification

Simplify $\sin^2 x + \sin^2 x \cot^2 x$.

Factor: $\sin^2 x (1 + \cot^2 x)$. Use $1 + \cot^2 x = \csc^2 x$: $\sin^2 x \cdot \csc^2 x$.

Since $\csc^2 x = \dfrac{1}{\sin^2 x}$:

$\sin^2 x \cdot \dfrac{1}{\sin^2 x} = 1$

Key takeaways

• $\sin^2 + \cos^2 = 1$ is the foundation; the other two Pythagoreans come from dividing it.
• $\tan = \sin/\cos$. Reciprocal pairs: $\sin/\csc$, $\cos/\sec$, $\tan/\cot$.
• $\sin(2\theta) = 2 \sin \theta \cos \theta$; $\cos(2\theta)$ has three useful forms.