Trig Identity Calculator

Prove, verify and simplify trigonometric identities, with each identity named as it's used.

Try an example:

What a trig identity is

An identity is an equation that holds for every value of the variable where both sides are defined. $sin^{2} x + cos^{2} x = 1$ is true for every $x$ , unlike an equation such as $sin x = \frac{1}{2}$ , which holds only for particular angles. Proving an identity means transforming one side into the other using identities you already know.

How to use the trig identity calculator

Type or paste your problem into the box above — for example Prove (1 - cos^2 x)/sin x = sin x. Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

The identities you'll use most

Pythagorean: $sin^{2} x + cos^{2} x = 1$ and $1 + tan^{2} x = sec^{2} x$
Quotient and reciprocal: $tan x = \frac{s i n x}{c o s x}$ and $sec x = \frac{1}{c o s x}$
Double angle: $sin 2 x = 2 sin x cos x$ and $cos 2 x = cos^{2} x - sin^{2} x = 1 - 2 sin^{2} x$
Sum and difference: $sin (a \pm b) = sin a cos b \pm cos a sin b$

A strategy for proofs

Start from the more complicated side and work towards the simpler one. Don't move terms across the equals sign — you're proving the identity, not solving it.
Rewrite everything in terms of $sin$ and $cos$ , and combine fractions over a common denominator.
Look for a Pythagorean identity or a difference of squares, like $1 - cos^{2} x$ , to simplify.

Worked example: prove $\frac{sin ^{2} x}{1 - cos x} = 1 + cos x$

Start with the left side $\frac{sin ^{2} x}{1 - cos x}$
Use the Pythagorean identity $sin^{2} x = 1 - cos^{2} x$ , so the left side is $\frac{1 - cos ^{2} x}{1 - cos x}$ .
Factor the difference of squares $\frac{( 1 - cos x ) ( 1 + cos x )}{1 - cos x}$
Cancel $= 1 + cos x$ , which is the right side. The identity holds wherever $cos x \neq = 1$ .

Common mistakes to avoid

Treating functions like variables

$sin 2 x$ is not $2 sin x$ , and $sin (a + b)$ is not $sin a + sin b$ .

Working on both sides at once

Cross-multiplying or moving terms assumes the identity is already true. Transform one side until it matches the other.

Ignoring where it's defined

Identities with fractions only hold where the denominator isn't zero, and it's worth saying so.

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Questions students ask

Can it prove any trig identity?

It works through the identities you'll meet in high school and college trigonometry, naming each one it uses. If an identity is false, it shows a value of $x$ where the two sides differ.

Can it solve trig equations too?

Yes. $2 sin^{2} x - 1 = 0$ on $0 \leq x < 2 π$ gives $x = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$ . Say which interval you want.

Degrees or radians?

Either. Say which you want in the question; otherwise it follows whatever the problem uses.

How do I type sin squared?

Type sin^2(x) or (sin x)^2. Both mean $sin^{2} x$ .