Trigonometric Identities

Take any angle — say \theta = 37°. Compute \sin^2 37° + \cos^2 37° on your calculator. You will get 1. Try \theta = 123°. Again, 1. Try \theta = -500°. Still 1.

This is not a coincidence, and it is not a rounding artifact. The equation \sin^2\theta + \cos^2\theta = 1 holds for every angle \theta, exactly, with no exceptions. That is what makes it an identity — it is not an equation to "solve" (there is nothing to solve for), but a permanent fact about sine and cosine that you can use anywhere, any time, without checking whether it applies.

The proof is one line, and it comes straight from the unit circle. The point at angle \theta on the unit circle has coordinates (\cos\theta, \sin\theta). This point lies on the circle x^2 + y^2 = 1. So:

That is the entire proof. The identity is the equation of the circle, read in terms of trigonometric functions.

Why is it called the Pythagorean identity? Because it is the Pythagorean theorem in disguise. Inside the unit circle, drop a perpendicular from the point (\cos\theta, \sin\theta) to the x-axis. You get a right triangle with hypotenuse 1 (the radius), horizontal leg \cos\theta, and vertical leg \sin\theta. The Pythagorean theorem says \text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2, which is exactly \cos^2\theta + \sin^2\theta = 1.

This identity is the single most-used fact in trigonometry. It appears in nearly every simplification, every proof, and every problem. If you remember one thing from this article, remember this.

The three Pythagorean identities

The identity \sin^2\theta + \cos^2\theta = 1 is the parent. The other two are obtained by dividing it through.

Divide by \cos^2\theta (valid whenever \cos\theta \neq 0):

Divide by \sin^2\theta (valid whenever \sin\theta \neq 0):

These three identities are the foundation of all trigonometric simplification.

The reciprocal and quotient identities

These are definitional — they follow directly from how the six functions are defined.

Reciprocal identities:

Quotient identities:

These are not deep — they are just the definitions written as equations. But they are used constantly, because converting everything to sine and cosine is often the first step in proving an identity.

The complete toolkit

Before diving into proof strategies, here is the full collection of identities you will use most often. The Pythagorean three are the foundation. The reciprocal and quotient identities are definitional. The co-function identities come from the unit circle:

And the even/odd identities encode the symmetry of the unit circle:

These are not new facts — they all follow from the unit-circle definitions you learned in Trigonometric Ratios of Any Angle. But collected here, they form the toolkit you will reach for every time you face an identity to prove.

Strategies for proving identities

A trigonometric identity is an equation that you are asked to prove, not solve. The standard approach: take one side (usually the more complicated one), transform it step by step using known identities, and arrive at the other side. You do not cross the equals sign — you work only on one side.

Here are the three most useful strategies.

Strategy 1: Convert everything to sin and cos

Replace \tan\theta, \sec\theta, \csc\theta, and \cot\theta with their definitions in terms of sine and cosine. Then simplify. This brute-force approach works surprisingly often.

Example. Prove that \sec\theta - \cos\theta = \sin\theta \cdot \tan\theta.

Start with the left side:

Now use the Pythagorean identity: 1 - \cos^2\theta = \sin^2\theta.

That is the right side. The identity is proved.

You can verify this graphically: plot both \sec\theta - \cos\theta and \sin\theta\tan\theta, and you will see a single curve — the two expressions produce identical outputs for every valid input.

Strategy 2: Use algebraic identities

The factoring patterns from algebra — a^2 - b^2 = (a-b)(a+b), (a+b)^2 = a^2 + 2ab + b^2, a^3 + b^3 = (a+b)(a^2 - ab + b^2) — apply to trigonometric expressions just as well as to ordinary polynomials.

Example. Simplify \dfrac{1 - \sin^2\theta}{1 + \sin\theta}.

The numerator is a difference of squares: 1 - \sin^2\theta = (1 - \sin\theta)(1 + \sin\theta).

The (1 + \sin\theta) cancels (provided \sin\theta \neq -1), leaving 1 - \sin\theta. You could also have recognised 1 - \sin^2\theta = \cos^2\theta and written \cos^2\theta / (1 + \sin\theta), which leads to the same answer after multiplying numerator and denominator by (1 - \sin\theta).

Strategy 3: Multiply by a conjugate

When a denominator contains 1 + \sin\theta or 1 - \cos\theta, multiply numerator and denominator by the conjugate. This turns the denominator into a Pythagorean form.

Example. Prove that \dfrac{1}{1 + \sin\theta} + \dfrac{1}{1 - \sin\theta} = 2\sec^2\theta.

Combine the two fractions on the left:

The denominator became \cos^2\theta via the Pythagorean identity. The sine terms in the numerator cancelled. The whole thing collapsed in one line.

Strategy 4: Work from the more complex side

This is not a separate algebraic trick — it is a general principle. When one side of an identity is clearly more complex than the other (more terms, more fractions, more nested functions), start from that side and simplify toward the simpler one. The reason: simplifying is easier than complicating. Going from \sec\theta - \cos\theta to \sin\theta\tan\theta is natural; going the other direction — expanding \sin\theta\tan\theta into something that looks like \sec\theta - \cos\theta — requires you to know the destination in advance.

If both sides are equally complex, a useful fallback: convert both sides independently to sine and cosine and check that they match.

Conditional identities

A conditional identity holds not for all angles, but under a specific condition — usually something like A + B + C = 180° (the angle-sum condition in a triangle). Under that constraint, you derive identities that would not be true otherwise.

Key fact for triangles. If A + B + C = 180°, then C = 180° - (A + B), which gives

using the allied angle identity \sin(180° - x) = \sin x and \cos(180° - x) = -\cos x.

Example. If A + B + C = 180°, prove that \sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C.

Start with the left side. Use the sum-to-product formula on the first two terms:

Since A + B = 180° - C, \sin(A + B) = \sin C. So:

Now deal with the third term. Write \sin 2C using the double-angle formula: \sin 2C = 2\sin C\cos C. Since C = 180° - (A + B), \cos C = -\cos(A + B). So:

Putting it all together:

Now use the identity \cos P - \cos Q = 2\sin\!\left(\frac{Q+P}{2}\right)\sin\!\left(\frac{Q-P}{2}\right). Here P = A - B and Q = A + B:

(since (Q+P)/2 = A and (Q-P)/2 = B).

So the expression becomes:

which is the right side. The identity is proved.

Maximum and minimum values

A frequent JEE problem type: find the maximum and minimum values of a trigonometric expression. The Pythagorean identity and the bounded range of sine and cosine (-1 \leq \sin\theta \leq 1, -1 \leq \cos\theta \leq 1) are the main tools.

The a\sin\theta + b\cos\theta form. The expression a\sin\theta + b\cos\theta can always be written as a single sinusoidal:

where \tan\phi = b/a. The proof uses the angle-addition formula:

Choose \cos\phi = a/\sqrt{a^2 + b^2} and \sin\phi = b/\sqrt{a^2 + b^2}. Then:

Since -1 \leq \sin(\theta + \phi) \leq 1, the expression a\sin\theta + b\cos\theta is bounded:

This is an extremely useful result. The maximum value of a\sin\theta + b\cos\theta is \sqrt{a^2 + b^2}, and the minimum is -\sqrt{a^2 + b^2}.

Example. Find the maximum value of 3\sin\theta + 4\cos\theta.

It occurs when \sin(\theta + \phi) = 1, where \tan\phi = 4/3. Since \phi = \arctan(4/3) \approx 53.13°, the maximum is achieved at \theta = 90° - 53.13° \approx 36.87°.

For expressions of the form a\sin^2\theta + b\cos^2\theta. Use \sin^2\theta + \cos^2\theta = 1 to write the expression in terms of just one squared function. Since \sin^2\theta and \cos^2\theta each range over [0, 1], and they always add to 1, the expression a\sin^2\theta + b\cos^2\theta is a weighted average and always lies between the smaller of a, b and the larger.

This is a quick way to find the range of many JEE-style expressions without any detailed algebra.

Two worked examples

Common confusions

• "\sin^2\theta + \cos^2\theta = 1 only works for acute angles." It works for every angle — that is the whole point of calling it an identity. The proof uses the equation of the unit circle, which has no restriction on \theta.

• "To prove an identity, start from both sides and meet in the middle." This is technically valid but risky — it can lead you to circular reasoning if you are not careful. The safer method is to start from one side only and transform it into the other.

• "\sin^2\theta means \sin(\sin\theta)." It means (\sin\theta)^2 — the sine of \theta, squared. This notation is unique to trigonometry. The superscript 2 sits between \sin and \theta to avoid the ambiguity of \sin\theta^2, which could mean \sin(\theta^2).

• "\sqrt{a^2 + b^2} is the maximum of a\sin\theta + b\cos\theta only when a and b are positive." The formula \sqrt{a^2 + b^2} gives the maximum regardless of the signs of a and b. The signs affect where the maximum occurs (the phase angle \phi), not what the maximum is.

• "Conditional identities are separate formulas to memorise." They are not standalone formulas — they are consequences of the fundamental identities applied under an extra constraint (like A + B + C = 180°). Understanding the derivation is more reliable than memorising the result.

Going deeper

The identity as a geometric invariant

The Pythagorean identity \sin^2\theta + \cos^2\theta = 1 says that no matter how \theta changes, the point (\cos\theta, \sin\theta) stays on the unit circle. In physics language, \sin^2\theta + \cos^2\theta is a conserved quantity — it does not change as \theta varies. Conserved quantities are powerful because they let you relate things at different stages of a process.

The same principle extends. The identity \cosh^2 x - \sinh^2 x = 1 (for hyperbolic functions) says that a point (\cosh x, \sinh x) stays on a hyperbola. Different curve, same idea: the identity is the equation of the curve.

Why identities feel different from equations

An equation like \sin\theta = 1/2 has specific solutions (\theta = 30°, 150°, 390°, \ldots). An identity like \sin^2\theta + \cos^2\theta = 1 has every angle as a solution. This is the defining difference: an equation constrains \theta; an identity does not. When you "prove" an identity, you are showing that the two sides are the same function — the same output for every possible input.

This is why the approach is different. For equations, you manipulate to isolate \theta. For identities, you manipulate to show that two expressions are algebraically equivalent. The goal is not a number — it is a logical demonstration.

Generating identities by differentiation

Here is a surprising connection. The Pythagorean identity can be differentiated to produce another identity. Start with \sin^2\theta + \cos^2\theta = 1. Differentiate both sides with respect to \theta:

This is trivially true — 0 = 0. But the intermediate expression 2\sin\theta\cos\theta is the double-angle formula for sine: \sin 2\theta = 2\sin\theta\cos\theta. The derivative of a constant is zero, and the structure of the derivative encodes the double-angle identity. This is not the standard way to derive \sin 2\theta, but it reveals a deep connection between identities and calculus.

What if you differentiate the second Pythagorean identity, 1 + \tan^2\theta = \sec^2\theta? The derivative of the left side is 2\tan\theta \cdot \sec^2\theta. The derivative of the right side is 2\sec\theta \cdot \sec\theta\tan\theta = 2\sec^2\theta\tan\theta. They match — confirming the identity by a different route.

How many identities are there?

In one sense, there are infinitely many — you can always create new ones by combining known identities. But they all trace back to a small set of generators. The Pythagorean identity, the angle-addition formulas (\sin(A + B) and \cos(A + B)), and the periodicity identity (\sin(\theta + 2\pi) = \sin\theta) together generate every standard trigonometric identity. Everything else — double-angle, half-angle, product-to-sum, sum-to-product, triple angle — is derived from these three building blocks.

Madhava and the Kerala school of mathematicians understood this generative structure in the 14th century, using it to derive infinite series for sine and cosine that anticipated the Taylor series by three centuries.

The a\sin\theta + b\cos\theta result and phasors

The result that a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \phi) has a beautiful geometric interpretation. Think of a and b as the components of a vector (a, b) in the plane. The length of that vector is \sqrt{a^2 + b^2}, and the angle it makes with the horizontal is \phi = \arctan(b/a). The expression a\sin\theta + b\cos\theta is the projection of this vector onto a rotating direction — and the maximum projection is the full length of the vector.

In electrical engineering, this interpretation is called a phasor. Alternating currents are described by expressions of the form a\sin(\omega t) + b\cos(\omega t), and the phasor \sqrt{a^2 + b^2} is the amplitude of the combined signal. The trigonometric identity is doing real engineering work.

Where this leads next

The identities you have learned here are the starting point for solving equations and proving more advanced results.

• Trigonometric Equations — using identities to find all angles that satisfy a given equation.
• Trigonometric Ratios of Any Angle — the unit-circle definitions that make these identities geometrically meaningful.
• Algebraic Identities — the algebraic factoring patterns (a^2 - b^2, a^3 + b^3, etc.) that you use as tools inside trig proofs.
• De Moivre's Theorem — the connection between trigonometric identities and complex numbers, where (\cos\theta + i\sin\theta)^n generates multi-angle formulas automatically.
• Trigonometric Ratios — a refresher on the basic definitions and standard angle values.