Trigonometric Identities: The Essential List and How to Use It

Identities versus equations

The equation $\sin x = \frac{1}{2}$ is true only for particular angles, so it is something to solve. The equation $\sin^2 x + \cos^2 x = 1$ is true for every angle, so it is an identity, something to use. Identities are tools for rewriting an expression in a more convenient form without changing its value.

You can always test a claimed identity numerically. If it fails at a single angle, it is not an identity. If it holds at several unrelated angles, it is probably true, but a proof is still needed to be certain. That numerical habit catches many sign errors and is worth using while you learn.

The basic identities

Reciprocal and quotient identities define the other four functions in terms of sine and cosine:

The Pythagorean identity comes from the equation of the unit circle, $x^2 + y^2 = 1$, with $x = \cos\theta$ and $y = \sin\theta$. Dividing it by $\cos^2 x$ or by $\sin^2 x$ gives two more versions:

Symmetry of the unit circle gives the even and odd identities, $\cos(-x) = \cos x$, $\sin(-x) = -\sin x$ and $\tan(-x) = -\tan x$, and the cofunction identities, such as $\sin\left(\frac{\pi}{2} - x\right) = \cos x$. The cofunction identities explain the names: cosine is the complement's sine. The unit circle guide shows where each of these comes from.

Sum and difference formulas

These formulas express the trig functions of a sum or difference of angles in terms of the separate angles. They are the source of almost every other identity.

Note the sign switch in the cosine formula: the cosine of a sum has a minus sign. A common mistake is to assume $\sin(A + B) = \sin A + \sin B$. Testing with $A = B = \frac{\pi}{2}$ shows it fails: the left side is $\sin\pi = 0$ but the right side is 2.

In the same way, $\tan 15^\circ = \tan(45^\circ - 30^\circ) = \frac{1 - \frac{\sqrt{3}}{3}}{1 + \frac{\sqrt{3}}{3}} = 2 - \sqrt{3} \approx 0.2679$.

Double angle and power reduction

Setting $B = A$ in the sum formulas gives the double angle identities. Cosine has three equivalent forms, obtained by replacing $\sin^2$ or $\cos^2$ using the Pythagorean identity.

Solving the last two forms for the squares gives the power reduction identities, which turn squares into first powers of a double angle. They are essential in calculus, where $\sin^2 x$ is hard to integrate directly.

For example, $\int \sin^2 x\,dx = \int \frac{1 - \cos 2x}{2}\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$. Replacing $x$ by $\frac{x}{2}$ in these identities gives the half-angle formulas, such as $\cos^2\frac{x}{2} = \frac{1 + \cos x}{2}$.

How to prove an identity

Proving an identity means transforming one side, step by step, until it becomes the other. Do not treat it as an equation by doing the same thing to both sides, because that assumes what you are trying to prove. These strategies work in most cases:

1. Start with the more complicated side; there is more to simplify.
2. Rewrite everything in terms of sine and cosine.
3. Look for Pythagorean patterns, such as $1 - \cos^2 x$, and replace them.
4. Combine fractions over a common denominator, or split one fraction into two.
5. Factor, or multiply by a conjugate such as $1 + \sin x$, when a sum appears in a denominator.
6. Keep the target side in view so each step moves towards it.

A shorter one combines two families: $(\sin x + \cos x)^2 = \sin^2 x + 2\sin x\cos x + \cos^2 x = 1 + \sin 2x$.

Using identities to solve equations

Identities often turn a trig equation into a quadratic in a single function. Solve the quadratic, then use the unit circle to find every angle.

Avoid dividing both sides by a trig function, such as $\sin x$, because you can lose the solutions where it equals zero. Move everything to one side and factor instead.

The factoring steps here are the same as for ordinary quadratics; see how to solve quadratic equations for a refresher.

Which identities to memorise

You do not need to memorise every formula separately. Learn the Pythagorean identity, the sum formulas for sine and cosine, and the definitions of the other four functions. Everything else can be derived in a line or two: double angle formulas from the sum formulas with $B = A$, the other Pythagorean forms by division, power reduction by solving the cosine double angle formula, and the difference formulas from the sum formulas using the even and odd identities.

Being able to derive a formula is more reliable under exam pressure than recalling it, because you can check it as you go. Many exams supply a formula sheet anyway, so the real skill being tested is choosing which identity to use, and that only comes with practice on a variety of problems.

Common mistakes

• Writing $\sin(A + B) = \sin A + \sin B$ or $\sin 2x = 2\sin x$.
• Getting the sign wrong in $\cos(A + B)$, which has a minus sign.
• Confusing $\sin^2 x$, meaning $(\sin x)^2$, with $\sin(x^2)$.
• Working on both sides of an identity at once as if it were an equation.
• Dividing an equation by a trig function and losing solutions.

The trigonometric identity calculator checks whether two expressions are identical and shows a simplification path, which is a good way to test your own proofs.

Where identities show up later

Identities are not just an exercise for a trigonometry chapter. In calculus, the derivative of sine is found with the sum formula and two limits, integrals of powers of sine and cosine rely on power reduction, and trigonometric substitution uses $1 + \tan^2 x = \sec^2 x$ to remove square roots such as $\sqrt{1 + x^2}$.

In physics and engineering, adding two waves of the same frequency gives another wave of that frequency, and the sum formula shows exactly how. Writing $a\sin x + b\cos x$ as $R\sin(x + \varphi)$, with $R = \sqrt{a^2 + b^2}$, tells you the amplitude of the combined wave immediately. For example, $3\sin x + 4\cos x$ has amplitude 5, so its largest value is 5 and its smallest is $-5$. Expanding $5\sin(x + \varphi)$ with the sum formula and matching coefficients gives $\cos\varphi = \frac{3}{5}$ and $\sin\varphi = \frac{4}{5}$.

Signal processing, computer graphics rotations and the laws used to solve triangles all rest on the same small set of formulas, which is why learning them well pays off for years, long after the trigonometry unit itself has finished.

Frequently asked questions