Trigonometric Identities: Core List, Proof Ideas, and Example

Trigonometric identities are equations involving $\sin$, $\cos$, $\tan$, and related functions that hold for every angle where both sides are defined. They are the rewriting tools behind simplifying expressions, solving trig equations, and setting up integrals. The fastest way to make them stick is to group them by purpose: some rewrite one function in terms of another, some link $\sin \theta$ and $\cos \theta$, and some change the angle from $\theta$ to $2\theta$ or $\theta/2$.

One distinction first, because it controls everything else. An identity is true for every angle in its domain, like

$$\sin^2 \theta + \cos^2 \theta = 1$$

By contrast,

$$\sin \theta = \frac{1}{2}$$

is not an identity; it is true only for specific angles. And the domain condition is part of the identity. For instance,

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

holds only when $\cos \theta \neq 0$.

The Identities And Their Symbols

Reciprocal identities

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

Each formula requires the denominator to be nonzero.

Quotient identities

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

These are often the first step in simplification problems because they rewrite everything in terms of $\sin$ and $\cos$.

Pythagorean identities

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

The first identity is the source of the other two.

Even-odd identities

$$\sin(-\theta) = -\sin \theta, \qquad \cos(-\theta) = \cos \theta, \qquad \tan(-\theta) = -\tan \theta$$

The same pattern extends to the reciprocal functions: $\csc$ and $\cot$ are odd, while $\sec$ is even.

Cofunction identities

$$\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta$$ $$\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta$$ $$\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta$$

These come from complementary angles.

Sum and difference identities

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$ $$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$ $$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$ $$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$ $$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$ $$\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$$

For the tangent formulas, the denominator must be nonzero.

Double-angle identities

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

The tangent version also needs $1 - \tan^2 \theta \neq 0$.

Half-angle identities

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

For an angle written as $\theta/2$, the square-root forms are

$$\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$$ $$\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$$

The sign depends on the quadrant of $\theta/2$, so the $\pm$ cannot be dropped blindly.

Why The Main Identities Hold

The unit circle gives the first Pythagorean identity. On the unit circle, the point at angle $\theta$ is $(\cos \theta, \sin \theta)$. Because every point on that circle satisfies $x^2 + y^2 = 1$, substituting $x = \cos \theta$ and $y = \sin \theta$ gives

$$\cos^2 \theta + \sin^2 \theta = 1$$

The other two Pythagorean identities come from division. If $\cos \theta \neq 0$, divide $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$:

$$\frac{\sin^2 \theta}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta}$$ $$\tan^2 \theta + 1 = \sec^2 \theta$$

If $\sin \theta \neq 0$, dividing by $\sin^2 \theta$ gives $1 + \cot^2 \theta = \csc^2 \theta$.

The double-angle identities come from the angle-sum formulas. Start with

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

and set $\alpha = \beta = \theta$:

$$\sin(2\theta) = 2\sin \theta \cos \theta$$

The cosine and tangent double-angle identities are derived the same way.

Worked Example: Simplify A Double-Angle Expression

Simplify

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

for angles where the original expression is defined. Use the double-angle identities:

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

and

$$\sin(2\theta) = 2\sin \theta \cos \theta$$

Now substitute:

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

This is valid only where the original denominator is nonzero, so $\sin(2\theta) \neq 0$. That condition matters because canceling a factor can hide values that were excluded at the start.

Now You Try

Simplify

$$\frac{\sin(2\theta)}{1 + \cos(2\theta)}$$

using double-angle identities, and keep the domain condition from the original expression in view. Work it slowly: substitute $\sin(2\theta)=2\sin\theta\cos\theta$ and $1+\cos(2\theta)=2\cos^2\theta$, then cancel. Your result should again be $\tan \theta$; if it is, compare the two problems to see how the same identities reduce different-looking expressions to the same answer.

Calculation Traps To Watch

Ignoring domain restrictions causes the most trouble. Dividing by $\sin \theta$ or $\cos \theta$ is valid only when that quantity is not zero. Dropping the $\pm$ in half-angle formulas is the next most common error; the square root alone does not determine the sign of the trig value. Students also mix up $\sin^2 \theta$ and $\sin(\theta^2)$ — the notation $\sin^2 \theta$ means $(\sin \theta)^2$.

When Trig Identities Are Used

Trig identities show up whenever you need to rewrite an expression into a more useful form: simplifying homework, proving two expressions are equal, solving trig equations, and preparing for calculus topics such as integration. In practice, many problems become easier once everything is rewritten in terms of $\sin \theta$ and $\cos \theta$.