Verifying Trig Identities

Verifying trig identities means proving that two trig expressions are equal for every angle where both are defined. Start with the more complicated side and rewrite it, one step at a time, until it matches the other side. Useful moves are rewriting in sine and cosine, using Pythagorean identities, combining fractions and factoring.

Before this lesson: You should know Fundamental Trig Identities, Adding and Subtracting Rational Expressions and Special Products of Binomials.

A Proof, Not an Equation to Solve

To verify an identity, you do not solve for \theta, and you do not do the same thing to both sides. That would assume the identity is already true. Instead, you work on one side only and change it into the other side. Each step must use algebra or an identity you already know.

It helps to work like a mechanic who knows what the finished part should look like. Look at the target side. If it has only sine and cosine, rewrite your side in sine and cosine. If it has no fractions, look for a way to combine and cancel.

Strategies for Verifying Identities

1. Start with the more complicated side.
2. Rewrite tangent, cotangent, secant and cosecant in terms of sine and cosine.
3. Combine fractions over a common denominator, or split a fraction into two.
4. Expand products or factor expressions, such as a difference of squares.
5. Replace patterns like 1 - \sin^2 \theta with \cos^2 \theta using a Pythagorean identity.
6. If there is 1 - \cos \theta or 1 + \sin \theta in a denominator, try multiplying by its conjugate.
7. Stop when your side looks exactly like the other side.

Worked Examples

Example 1: Rewrite in sine and cosine

Verify \tan \theta \cos \theta = \sin \theta.

\tan \theta \cos \theta

\frac{\sin \theta}{\cos \theta} \cdot \cos \theta

\sin \theta

This matches the right side.

Answer: verified, \tan \theta \cos \theta = \sin \theta

Example 2: Combine fractions

Verify \sec \theta - \cos \theta = \sin \theta \tan \theta.

\sec \theta - \cos \theta

\frac{1}{\cos \theta} - \cos \theta

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

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

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

\sin \theta \cdot \frac{\sin \theta}{\cos \theta}

\sin \theta \tan \theta

Answer: verified. The left side became the right side.

Example 3: Expand a product

Verify (1 + \sin \theta)(1 - \sin \theta) = \cos^2 \theta.

(1 + \sin \theta)(1 - \sin \theta)

1 - \sin^2 \theta

\cos^2 \theta

Answer: verified, (1 + \sin \theta)(1 - \sin \theta) = \cos^2 \theta

Example 4: Multiply by a conjugate

Verify \frac{\sin \theta}{1 - \cos \theta} = \frac{1 + \cos \theta}{\sin \theta}.

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

\frac{\sin \theta (1 + \cos \theta)}{(1 - \cos \theta)(1 + \cos \theta)}

\frac{\sin \theta (1 + \cos \theta)}{1 - \cos^2 \theta}

\frac{\sin \theta (1 + \cos \theta)}{\sin^2 \theta}

\frac{1 + \cos \theta}{\sin \theta}

Answer: verified. Multiplying by the conjugate is the same trick used in rationalizing the denominator.

Example 5: Two answers on a test

On a test, Ana simplified an expression to \csc \theta - \sin \theta, and Ben got \cot \theta \cos \theta. Show that both answers are correct by verifying \csc \theta - \sin \theta = \cot \theta \cos \theta.

\csc \theta - \sin \theta

\frac{1}{\sin \theta} - \sin \theta

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

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

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

\frac{\cos \theta}{\sin \theta} \cdot \cos \theta

\cot \theta \cos \theta

Answer: Ana’s expression becomes Ben’s, so both answers are correct.

Common Mistakes

Related Methods

Work on both sides separately

Some teachers allow you to simplify each side on its own until both reach the same expression. This is fine as long as you never move a term from one side to the other.

Check with a graph

Graph both sides on Desmos or a calculator. If the graphs are exactly the same, the identity is probably true. If they differ anywhere, it is not an identity.

Factor like a polynomial

Treat \sin \theta and \cos \theta like variables. For example, \sin^4 \theta - \cos^4 \theta factors as a difference of squares: (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta).

Practice Problems

1. Verify \sin \theta \cot \theta = \cos \theta.
2. Verify \cos \theta \sec \theta = 1.
3. Verify (1 - \cos \theta)(1 + \cos \theta) = \sin^2 \theta.
4. Verify \sec^2 \theta - \tan^2 \theta = 1.
5. Verify \tan \theta + \cot \theta = \sec \theta \csc \theta.
6. Verify \csc^2 \theta (1 - \cos^2 \theta) = 1.
7. Verify \frac{\cos \theta}{1 - \sin \theta} = \frac{1 + \sin \theta}{\cos \theta}.
8. Verify (\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \cos \theta.
9. Verify \sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta.
10. A carpenter’s roof formula uses (\sec \theta - 1)(\sec \theta + 1). Show that it equals \tan^2 \theta.

\sin \theta \cdot \frac{\cos \theta}{\sin \theta}

\cos \theta

\cos \theta \cdot \frac{1}{\cos \theta}

1

1 - \cos^2 \theta

\sin^2 \theta

1 + \tan^2 \theta - \tan^2 \theta

1

\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}

\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}

\frac{1}{\sin \theta \cos \theta}

\frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} = \sec \theta \csc \theta

\csc^2 \theta \cdot \sin^2 \theta

\frac{1}{\sin^2 \theta} \cdot \sin^2 \theta

1

\frac{\cos \theta (1 + \sin \theta)}{1 - \sin^2 \theta}

\frac{\cos \theta (1 + \sin \theta)}{\cos^2 \theta}

\frac{1 + \sin \theta}{\cos \theta}

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

(\sin^2 \theta + \cos^2 \theta) + 2 \sin \theta \cos \theta

1 + 2 \sin \theta \cos \theta

(\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)

(\sin^2 \theta - \cos^2 \theta) \cdot 1

\sin^2 \theta - \cos^2 \theta

\sec^2 \theta - 1

1 + \tan^2 \theta - 1

\tan^2 \theta

FAQ

How do you verify a trig identity?

Pick one side, usually the more complicated one. Rewrite it step by step, using algebra and known identities, until it matches the other side exactly.

Can you work on both sides of an identity?

You can simplify each side separately to the same expression, but you cannot add, multiply or move terms across the equals sign. That would assume what you are trying to prove.

What is the first step when verifying identities?

A good first step is to rewrite everything in terms of sine and cosine. It often shows which Pythagorean identity or common denominator to use.

How do you know if an equation is not an identity?

Test an angle. If the two sides give different values for even one angle where both are defined, the equation is not an identity.

Summary

• Verify an identity by transforming one side into the other.
• Start with the more complicated side.
• Rewrite in sine and cosine, combine fractions, expand or factor.
• Use Pythagorean identities to replace patterns like 1 - \sin^2 \theta.
• Multiply by a conjugate when 1 \pm \sin \theta or 1 \pm \cos \theta is in a denominator.

Previous lesson: Fundamental Trig Identities

Next lesson: Sum and Difference Formulas

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