Trigonometric Identities

Trigonometry connects the angles of a triangle to the lengths of its sides. Three ratios named sine, cosine, and tangent do most of the work.

Trigonometric Identities: the central idea

A trigonometric identity is an equation true for every input where both sides are defined. Proving one means transforming expressions, not solving for selected angles.

Words you need

Identity

An identity is an equation that stays true for every value of the variable where both sides are defined, so it describes a permanent relationship rather than a question with an answer.

Conditional equation

A conditional equation is one that holds only for certain inputs, such as $\sin x=1/2$, and it is the kind you solve for a list of angles rather than prove.

Pythagorean identity

A Pythagorean identity is one of the three statements $\sin^2x+\cos^2x=1$, $\tan^2x+1=\sec^2x$, and $1+\cot^2x=\csc^2x$, all of which come from the Pythagorean theorem applied to the unit circle.

Reciprocal identity

A reciprocal identity pairs a function with its flipped fraction, giving $\csc x=1/\sin x$, $\sec x=1/\cos x$, and $\cot x=1/\tan x$.

Quotient identity

A quotient identity expresses tangent and cotangent as fractions of sine and cosine, namely $\tan x=\sin x/\cos x$ and $\cot x=\cos x/\sin x$.

Domain restriction

A domain restriction is the list of inputs an expression must avoid because they would make a denominator zero, such as $\cos x\ne0$ whenever $\tan x$ or $\sec x$ appears.

What to know before this lesson

Know unit-circle definitions, reciprocal and quotient identities, algebraic fractions, and domain restrictions.

If one of those prerequisites is uncertain, use the Trigonometry subject guide to locate the earlier concept before memorizing a procedure.

Trigonometric identities: a worked example

Every step, with the arithmetic

1. Step 1 - Choose a side and state the goalProve $\dfrac{\cos x}{1-\sin x}=\sec x+\tan x$. The left side looks messier, so start there.
2. Step 2 - Multiply top and bottom by the conjugate$\dfrac{\cos x}{1-\sin x}\cdot\dfrac{1+\sin x}{1+\sin x}=\dfrac{\cos x(1+\sin x)}{(1-\sin x)(1+\sin x)}$
3. Step 3 - Expand the denominator$(1-\sin x)(1+\sin x)=1-\sin^2x$
4. Step 4 - Swap in the Pythagorean identity$1-\sin^2x=\cos^2x$, so the expression is $\dfrac{\cos x(1+\sin x)}{\cos^2x}$.
5. Step 5 - Cancel one cosine$\dfrac{\cos x(1+\sin x)}{\cos^2x}=\dfrac{1+\sin x}{\cos x}$
6. Step 6 - Split the single fraction into two$\dfrac{1+\sin x}{\cos x}=\dfrac{1}{\cos x}+\dfrac{\sin x}{\cos x}$
7. Step 7 - Name the two pieces$\dfrac{1}{\cos x}=\sec x$ and $\dfrac{\sin x}{\cos x}=\tan x$, so the left side equals $\sec x+\tan x$. Proof complete.
8. Step 8 - Record the excluded inputsThe original fraction needs $\sin x\ne1$ and the answer needs $\cos x\ne0$, so $x\ne\dfrac{\pi}{2}+\pi n$.

Dividing $\sin^2x+\cos^2x=1$ by $\cos^2x$ gives $\tan^2x+1=\sec^2x$ wherever $\cos x\ne0$.

The identities you reach for most, each checked numerically at x = 30 degrees

The step-by-step method for trigonometric identities

1. Work on one side—usually the more complicated side—and rewrite functions using sine and cosine when useful.
2. Apply algebraic factoring, common denominators, or a Pythagorean identity while recording excluded inputs.
3. Stop when the transformed side exactly matches the other side; do not cancel terms across addition.

How to check your answer

Substitute several valid angles as a diagnostic check, then inspect whether every transformation was reversible on the stated domain.

Confirm the quadrant and sign, convert degrees and radians consistently, and substitute solutions into the original interval because periodic equations usually have more than one answer.

A mistake that changes the mathematics

Checking three angles supports an identity but does not prove it. A proof must use known identities and valid algebra for the whole domain.

Where you will actually use this

Audio and beat frequencies

Two tones close in pitch, played together, produce a slow throbbing. The sum-to-product identity $\sin A+\sin B=2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}$ shows exactly why: the combined wave is one average tone whose volume is shaped by a slow cosine, and piano tuners listen for that throb.

Projectile range in physics

A projectile launched at speed $v$ and angle $\theta$ travels $R=\dfrac{v^2\sin\theta\cos\theta\cdot2}{g}$, which the double-angle identity collapses to $R=\dfrac{v^2\sin2\theta}{g}$. Since $\sin2\theta$ peaks at $2\theta=90^\circ$, the best launch angle is $45^\circ$.

Simplifying integrals

The integral of $\sin^2x$ has no obvious antiderivative until you rewrite it as $\dfrac{1-\cos2x}{2}$ using a double-angle identity. Then it integrates term by term to $\dfrac{x}{2}-\dfrac{\sin2x}{4}+C$.

How trigonometric identities connects to the rest of trigonometry

• The unit circle — Every identity here can be read off the unit circle, since $\sin^2x+\cos^2x=1$ is just the statement that the point $(\cos x,\sin x)$ sits one unit from the origin.
• Solving trig equations — Identities are the setup move for solving: rewriting $2\sin^2x+3\cos x=3$ with $\sin^2x=1-\cos^2x$ turns it into a quadratic in $\cos x$ that factors.
• Integration techniques — Trigonometric integrals are chosen and solved by identity, so knowing which substitution to make is really knowing which identity flattens the integrand.

Try a transfer problem

Prove $(1-\cos^2x)/\sin x=\sin x$ and state exactly which values the original expression excludes.

Show the worked answer

Start with the numerator. The Pythagorean identity $\sin^2x+\cos^2x=1$ rearranges to $1-\cos^2x=\sin^2x$, so the expression becomes $\dfrac{\sin^2x}{\sin x}$. One factor of $\sin x$ cancels against the denominator, leaving $\sin x$, which matches the right side. The proof is done. Now the domain: the original expression divides by $\sin x$, so it requires $\sin x\ne0$. Sine is zero at $0$, $\pi$, $2\pi$, and so on, so the excluded values are $x=\pi n$ for every whole number $n$. The simplified form $\sin x$ is happily defined at those points, which is exactly why the domain has to be stated separately: the two sides agree everywhere except at the holes.

Once you have an answer, check it with the trigonometry calculator. Work the problem yourself first: the calculator confirms the arithmetic, but choosing the method is the part that transfers to the next question.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to trigonometry practice questions.

Questions about trigonometric identities

What is the difference between proving an identity and solving an equation?

When you solve, you are hunting for the inputs that make a statement true, so you may add the same thing to both sides. When you prove an identity, the statement is already claimed true everywhere, so you transform one side alone until it looks like the other. Doing algebra to both sides assumes the very thing you are trying to show.

Can I work on both sides at the same time?

It is safer not to. If you must, every step has to be reversible, meaning you never multiply by something that could be zero and never square both sides. The cleanest habit is to rewrite the messier side only, or to reduce both sides separately to the same third expression.

Why do I have to state domain restrictions?

Because cancelling can quietly create a new expression that works at inputs where the original one did not. In $(1-\cos^2x)/\sin x=\sin x$, the right side is fine at $x=0$ but the left side divides by zero there. Naming the excluded inputs is what keeps the claim honest.

Do I really need to memorize all of them?

No. Memorize $\sin^2x+\cos^2x=1$, the quotient identities, and the reciprocal identities, then derive the rest. Divide the first one by $\sin^2x$ and you get $1+\cot^2x=\csc^2x$ in a single line; divide by $\cos^2x$ instead and you get $\tan^2x+1=\sec^2x$.