What is the difference between an identity and an equation?

An identity is true for every value of the variable, so there is nothing to solve — you use it to rewrite expressions. An equation like \sin x = 0.5 is true only for particular values, and the task is to find them.

Which identity should I try first?

The Pythagorean identity \sin^2\theta + \cos^2\theta = 1, in whichever direction helps — collapsing a squared pair to 1, or expanding a 1 into squares. Failing that, convert everything to sine and cosine and simplify from there.

Can I work on both sides of a "verify" problem?

Not by moving terms across the equals sign. You may simplify each side independently down to a common third expression and then state they are equal — but never apply the same operation to both sides, since that presumes the identity holds.

Do I have to memorise all of them?

The Pythagorean, reciprocal and quotient identities, yes — they come up constantly. The double-angle and sum formulas are worth knowing but are also derivable: every double-angle formula falls out of the sum formula with A = B.

When do I multiply by the conjugate?

When a denominator looks like 1 \pm \sin\theta or 1 \pm \cos\theta. Multiplying top and bottom by the conjugate creates a difference of squares that the Pythagorean identity immediately simplifies.

Why do identities matter in calculus?

Because a great many integrals are unsolvable in their given form and routine after one rewrite. Turning \sin^2 x into \frac{1 - \cos 2x}{2}, or clearing a \sqrt{1 - x^2} with a trig substitution, is standard practice.

You can start simplifying and verifying trig identities for free by adding the Chrome extension — try it on your own worksheet before deciding if you need anything more.

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Verify and Simplify Trig Identities Step by Step

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Photo Math Solver Solved

\sin^2\theta + \cos^2\theta + 4

1The first two terms are the Pythagorean identity, so they collapse to $1$ .
2The expression becomes $1 + 4 = 5$ — a constant, independent of $\theta$ .

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What are trig identities?

A trigonometric identity is an equation that holds for every value of the variable, not just a select few. $\sin^2\theta + \cos^2\theta = 1$ is true for every angle $\theta$ — that is what makes it an identity rather than an equation to solve.

The distinction matters for what you do with it. An equation like $\sin x = 0.5$ has specific solutions to find. An identity has nothing to solve; you either use it to rewrite an expression into a simpler form, or you verify that a claimed identity really is one.

Identities are the algebra of trigonometry. They let you replace one expression with an equal but more useful one — turning a product into a sum, a double angle into single angles, or a mess of secants and tangents into a single number. Almost every trig simplification, and a great deal of calculus, runs on them.

Where you'll actually use this

Signal processing: sum-to-product identities explain the beat frequencies heard when two tones interfere.
AC electrical engineering, where phase shifts are combined using the sum and difference formulas.
Simplifying integrands in calculus — $\sin^2 x$ becomes integrable via the double-angle identity.
Computer graphics: rotation matrices are built from the angle-addition formulas.
Physics of waves and oscillation, where superposition is an identity applied to two sinusoids.

Identity: An equation true for every value of the variable, used for rewriting rather than solving.
Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$ , and its variants for tangent and cotangent.
Reciprocal functions: Cosecant, secant and cotangent — the reciprocals of sine, cosine and tangent.
Conjugate: The partner expression $1 + \cos\theta$ to $1 - \cos\theta$ , used to clear denominators.
Double-angle formula: An identity expressing $\sin 2\theta$ or $\cos 2\theta$ in terms of functions of $\theta$ alone.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Pythagorean identity

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

The one to try first — also useful rearranged as $\sin^2\theta = 1 - \cos^2\theta$.

Pythagorean variants

1 + \tan^2\theta = \sec^2\theta, \quad 1 + \cot^2\theta = \csc^2\theta

The main identity divided through by $\cos^2\theta$ and $\sin^2\theta$.

Reciprocal identities

\csc\theta = \tfrac{1}{\sin\theta},\; \sec\theta = \tfrac{1}{\cos\theta},\; \cot\theta = \tfrac{1}{\tan\theta}

Note the mismatch: secant pairs with cosine, cosecant with sine.

Quotient identities

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

The route for converting everything to sine and cosine.

Double angle

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

Cosine has two more forms: $1 - 2\sin^2\theta$ and $2\cos^2\theta - 1$.

Sum and difference

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

And $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$ — note the flipped sign.

Even and odd

\cos(-\theta) = \cos\theta, \quad \sin(-\theta) = -\sin\theta

Cosine is even, sine and tangent are odd.

03Methods

How to solve trig identities

Pick the method that fits the problem in front of you.

When to use it: The expression is cluttered with

\tan

\sec

\csc

\cot

and no structure is obvious.

1Replace every function using the quotient and reciprocal identities.
2Combine the resulting fractions over a common denominator.
3Simplify the compound fraction and cancel common factors.
4Look for $\sin^2 + \cos^2$ appearing, which collapses to $1$ .

04Worked examples

Trig Identities, solved step by step

From easy to hard — pick a problem to see its full solution.

\sin^2\theta + \cos^2\theta + 4

1The first two terms are the Pythagorean identity, so they collapse to $1$ .
2The expression becomes $1 + 4 = 5$ — a constant, independent of $\theta$ .

Try it on your own trig identities homework

Photo Math Solver reads a trigonometric identity straight from a screenshot — a "verify that" question, an expression to simplify, a proof from a problem set — and shows each move with the identity that justified it. No staring at $\frac{\sin^2\theta}{1 - \cos\theta}$ wondering which substitution unlocks it.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

\cos\theta\sec\theta

Problem 2

\frac{1 - \sin^2\theta}{\cos\theta}

Problem 3

\text{Verify: } \cot\theta\sin\theta = \cos\theta

Problem 4

\sec^2\theta - \tan^2\theta

Problem 5

\text{Simplify } \frac{\cos 2\theta + 1}{\cos\theta}

Stuck on your trig identities homework? Screenshot it

Identity problems are dense with superscripts and stacked fractions, which makes them slow and error-prone to retype. Screenshot the expression with Photo Math Solver and the solution names the identity used at every line, so you can see which substitution unlocked it rather than just the final form.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Treating a "verify" question as an equation and moving terms across the equals sign.	Every previous chapter rewarded doing the same thing to both sides, so it is the trained reflex.	Work one side only until it matches the other. Operating on both sides assumes the identity is true, which is exactly what you are being asked to establish.
Writing $\sin^2\theta$ as $\sin(\theta^2)$ , or cancelling the $\sin$ in $\frac{\sin 2\theta}{\sin\theta}$ to get $2$ .	$\sin$ looks like a variable multiplying $\theta$ , so the usual algebraic cancellation seems to apply.	$\sin$ is a function, not a factor. $\sin^2\theta$ means $(\sin\theta)^2$ , and $\sin 2\theta$ must be expanded to $2\sin\theta\cos\theta$ before anything can cancel.
Pairing secant with sine and cosecant with cosine.	The initial letters suggest it — "s" with "s", "c" with "c" — but the actual pairing is the opposite.	$\sec = \frac{1}{\cos}$ and $\csc = \frac{1}{\sin}$ . The third letter is the reliable hook: the third letter of "secant" is the "c" of "cosine".
Using $\cos 2\theta = \cos^2\theta + \sin^2\theta$ instead of the minus version.	The Pythagorean identity is so familiar that its plus sign gets carried over to the double-angle formula.	It is $\cos 2\theta = \cos^2\theta - \sin^2\theta$ . The plus version would just equal $1$ for every angle, which is obviously not what $\cos 2\theta$ does.
Starting a verification from the simpler side and getting stuck.	The short side looks like the easier place to begin.	Start from the complicated side — there is more structure to break down and more identities that apply. Building complexity out of a bare $\sec\theta$ is guesswork.

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ