Trig Identity Calculator – Prove and Simplify Step by Step

About the Trigonometric Identity Calculator

Proving trigonometric identities is one of the most feared topics in precalculus. There are dozens of formulas to remember, no single obvious first move, and a proof can wander for half a page before it either closes or collapses. Students often know the identities perfectly well and still get stuck because they cannot see which one to apply next.

This calculator takes the guesswork out by following one reliable strategy every time. It rewrites both sides using only $\sin x$ and $\cos x$ , expands multiple angles such as $2x$ and $3x$ , moves everything onto one side over a common denominator, and then replaces every $\cos^2 x$ with $1 - \sin^2 x$ . If the numerator collapses to zero, the identity is proved; if it does not, the equation is not an identity, and the calculator shows a specific angle where the two sides disagree.

You can also enter an expression without an equals sign. The calculator then simplifies it with the same machinery and recognises standard results, turning $\tan x + \cot x$ into $\sec x \csc x$ , for example. The method is mechanical, which is exactly the point: once you see how it works you can apply it by hand to any identity in a textbook exercise set.

What is a trigonometric identity?

A trigonometric identity is an equation involving trigonometric functions that is true for every value of the variable where both sides are defined. $\sin^2 x + \cos^2 x = 1$ is an identity; $\sin x = \frac12$ is an equation that is true only for particular angles, so it is not.

The phrase "where both sides are defined" matters. $1 + \tan^2 x = \sec^2 x$ is an identity even though neither side exists at $x = 90°$ , because the equation holds at every angle where the expressions make sense. A proof therefore shows that the two sides agree wherever they are defined, not that they agree for every real number.

Identities are tools rather than goals. They are used to simplify expressions before integrating, to solve trigonometric equations, to rewrite waves in physics and engineering, and to turn products of sines and cosines into sums. Recognising when two different-looking expressions are secretly the same is a skill that pays off throughout calculus.

How to use the Trigonometric Identity Calculator

To prove an identity, type an equation with one equals sign, such as 1 + tan(x)^2 = sec(x)^2.
To simplify, type an expression with no equals sign, such as tan(x) + cot(x).
Use a single variable, normally $x$ . Powers can be typed as sin(x)^2 or as sin^2(x).
Multiple angles such as 2x and 3x, and shifts by multiples of pi/2 such as x + pi/2, are supported.
Press Simplify / prove and read the steps; for a false identity, note the counterexample angle.

Formulas

Pythagorean identity

\sin^2 x + \cos^2 x = 1

Quotient identities

\tan x = \frac{\sin x}{\cos x},\quad \cot x = \frac{\cos x}{\sin x}

Reciprocal identities

\sec x = \frac{1}{\cos x},\quad \csc x = \frac{1}{\sin x}

Double angle

\sin 2x = 2\sin x\cos x,\quad \cos 2x = \cos^2 x - \sin^2 x

Shift by a quarter turn

\sin\left(x + \tfrac{\pi}{2}\right) = \cos x,\quad \cos\left(x + \tfrac{\pi}{2}\right) = -\sin x

Worked example: Prove 1 + tan²x = sec²x

The worked example proves $1 + \tan^2 x = \sec^2 x$ . The left side becomes $1 + \frac{\sin^2 x}{\cos^2 x}$ and the right side becomes $\frac{1}{\cos^2 x}$ , so every function is now a sine or a cosine.

Subtracting the sides over the common denominator $\cos^2 x$ leaves the numerator $\cos^2 x + \sin^2 x - 1$ . Replacing $\cos^2 x$ with $1 - \sin^2 x$ makes every term cancel, so the numerator is identically zero and the identity holds wherever $\cos x \neq 0$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

1+{\tan\left( x\right)}^{2} \equiv {\sec\left( x\right)}^{2}

How we read your problem

Prove a trigonometric identity

1+{\tan\left( x\right)}^{2} = {\sec\left( x\right)}^{2}

Step-by-step solution

Rewrite everything in terms of sin x and cos x
Identities used:
$\tan\theta = \frac{\sin\theta}{\cos\theta},\quad \sec\theta = \frac{1}{\cos\theta}$
Left side in sine and cosine
$\frac{-1}{\sin^{2} x - 1}$
Right side in sine and cosine
$\frac{-1}{\sin^{2} x - 1}$
Subtract the sides and write the difference over a common denominator
The identity is true exactly when this numerator is zero for every value of the variable.
$\text{LHS} - \text{RHS} = \frac{N}{D},\quad N = \cos^{2} x + \sin^{2} x - 1$
Replace every cos²x with 1 − sin²x and collect terms
$N = 0$
The numerator is identically zero
Every term cancelled after using sin² + cos² = 1, so both sides are equal wherever they are defined.

Formulas used

Pythagorean identity
$\sin^2\theta + \cos^2\theta = 1$
Quotient and reciprocal identities
$\tan\theta = \tfrac{\sin\theta}{\cos\theta},\ \sec\theta = \tfrac{1}{\cos\theta},\ \csc\theta = \tfrac{1}{\sin\theta}$
Double angle
$\sin 2\theta = 2\sin\theta\cos\theta,\ \cos 2\theta = \cos^2\theta - \sin^2\theta$

After rewriting both sides with sine and cosine and replacing cos² with 1 − sin², the difference of the two sides is exactly 0, so 1 + tan(x)^2 = sec(x)^2 holds for every x at which both sides are defined.

Verification

Numeric spot check at 10 sample values: both sides agree.

Detailed explanation

The sine-and-cosine strategy

Every trigonometric function can be written in terms of sine and cosine, so converting everything to those two functions puts both sides of an identity in a common language. Tangent, cotangent, secant and cosecant disappear into fractions, and the problem becomes a question about rational expressions in $\sin x$ and $\cos x$ .

Once both sides are fractions, subtracting them and combining over a common denominator turns the identity into a single statement: a certain numerator must be zero. Denominators do not need to be examined further, because they only record where the original expressions were undefined.

The last obstacle is that $\sin x$ and $\cos x$ are not independent; they are linked by $\sin^2 x + \cos^2 x = 1$ . Replacing every even power of cosine with powers of $1 - \sin^2 x$ removes that link from even powers. After collecting like terms, a true identity leaves nothing behind. This is why the method always terminates with a clear answer for the expressions it accepts, unlike the trial-and-error approach of choosing one side and transforming it step by step.

Multiple angles and shifts

Angles such as $2x$ and $3x$ are first expanded into functions of $x$ itself. The double-angle formulas give $\sin 2x$ and $\cos 2x$ , and repeated use of the addition formulas gives $\sin 3x = 3\sin x - 4\sin^3 x$ . The calculator verifies that identity, along with $\cos 2x = 1 - 2\sin^2 x$ , using exactly the same pipeline as any other input.

Shifts by a multiple of $\frac{\pi}{2}$ are handled with the co-function and quarter-turn rules. In general the calculator accepts any angle of the form $kx + m\cdot\frac{\pi}{2}$ with whole numbers $k$ and $m$ . Half angles such as $\frac{x}{2}$ and arbitrary shifts such as $x + 1$ cannot be reduced to polynomials in $\sin x$ and $\cos x$ , so the calculator declines them with an explanation instead of producing an unreliable proof.

The calculator works with one variable. An addition formula such as $\sin(x + y) = \sin x\cos y + \cos x\sin y$ involves two independent angles and is outside its scope; it will say so rather than attempt it.

Disproving an equation

If the numerator does not reduce to zero, the equation is not an identity. To make this concrete, the calculator evaluates both sides at a sample angle and shows that they differ. For the common mistake $\sin 2x = 2\sin x$ , it reports that at $x = 30°$ the left side is about $0.866$ while the right side is $1$ .

A single counterexample is enough to disprove an identity, but no number of agreeing examples can prove one. Two expressions can match at many angles by coincidence. That is why the calculator's proof is algebraic, and why checking a few values on a graph is useful as a sanity check but not a substitute for a proof.

Simplifying expressions

Without an equals sign, the calculator rewrites the expression in sine and cosine, combines it into a single fraction, uses the Pythagorean identity and cancels common factors. It then compares the result with standard forms. For $\tan x + \cot x$ , the single fraction $\frac{\sin^2 x + \cos^2 x}{\sin x\cos x}$ reduces to $\frac{1}{\sin x\cos x}$ , which it recognises as $\sec x\csc x$ .

There is rarely one correct simplified form in trigonometry. $\sec x\csc x$ , $\frac{1}{\sin x\cos x}$ and $\frac{2}{\sin 2x}$ are all equivalent. The calculator chooses a compact standard form, and the steps show the intermediate fractions so you can stop at whichever version your course expects.

Writing a proof by hand

Teachers often require that you transform one side into the other rather than subtracting the sides, because working on both sides at once can hide invalid steps such as squaring. The calculator's steps translate directly into that style. Start with the more complicated side, rewrite it in sine and cosine, combine fractions, apply $\sin^2 x + \cos^2 x = 1$ in whichever direction helps, and simplify until it matches the other side.

Useful moves include multiplying by a conjugate, as in proving $\frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x}$ , factoring a difference of squares, as in $\sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x)$ , and splitting a fraction into separate terms. The calculator confirms that both of those identities are true, so you can focus on presenting the argument neatly.

Common mistakes

Treating $\sin 2x$ as $2\sin x$ ; the double-angle formula is $2\sin x\cos x$ .
Writing $\sin(a + b) = \sin a + \sin b$ ; trigonometric functions are not additive.
Cancelling across a sum, such as reducing $\frac{\sin x + \cos x}{\sin x}$ to $1 + \cos x$ .
Squaring both sides during a proof, which can turn a false statement into a true one.
Checking a few angles and concluding the identity is proved.

Tips

Start with the more complicated side; there is more to simplify.
Convert everything to sine and cosine when no other move is obvious.
Look for $\sin^2 x + \cos^2 x$ hiding in a numerator after combining fractions.
Multiply by a conjugate when you see $1 \pm \sin x$ or $1 \pm \cos x$ in a denominator.
Test the identity at one angle, such as $30°$ , before spending time on a proof.
Use the unit circle calculator to evaluate both sides at special angles quickly.

Frequently asked questions

How does the calculator prove an identity?

It rewrites both sides in sine and cosine, subtracts them over a common denominator and replaces cos²x with 1 − sin²x. If the numerator becomes zero, the identity is true wherever both sides are defined.

What happens if my equation is not an identity?

The calculator says so and gives a counterexample, an angle where the two sides have different values, such as x = 30° for sin 2x = 2 sin x.

Can it simplify a trig expression without an equals sign?

Yes. Enter just the expression. It is combined into one fraction, reduced with sin²x + cos²x = 1 and matched to a standard form, for example tan x + cot x becomes sec x csc x.

Which angles are supported?

Angles of the form kx plus a multiple of π/2, with k a whole number, such as 2x, 3x or x + π/2. Half angles like x/2 and expressions with two variables are not supported, and the calculator explains why.

Why does the answer say where both sides are defined?

Some identities involve functions that are undefined at certain angles, such as tan x at 90°. An identity only needs to hold where both sides exist.

Can it prove sin(x + y) = sin x cos y + cos x sin y?

No. That formula has two independent variables, and the calculator works with a single variable. It reports this rather than attempting a proof.

How should I type squared trig functions?

Either sin(x)^2 or the textbook form sin^2(x) works; the calculator reads both as the square of sin x. The same applies to the other five functions.

Is checking the graph enough to prove an identity?

No. Matching graphs strongly suggest an identity but cannot prove one. An algebraic argument like the one the calculator shows is needed; one mismatch, however, is enough to disprove it.

Prove or simplify your identity

Type a trigonometric equation or expression above to see a complete sine-and-cosine proof or simplification.

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