9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

Trigonometric identities are equations that hold true for every angle where both sides are defined. Mastering them gives you the tools to simplify complex expressions and verify equations, which comes up constantly in calculus and other advanced courses.

This guide covers the three main identity families (Pythagorean, reciprocal, and quotient), how to verify that an equation is an identity, and techniques for simplifying trigonometric expressions.

Verifying Trigonometric Identities

Application of fundamental trigonometric identities

A trigonometric identity is an equation that's true for all values of the variable where both sides are defined. For example, $\sin^2\theta + \cos^2\theta = 1$ holds for every $\theta$. This is different from a trigonometric equation, which is only true for specific values.

There are three families of identities you need to know cold:

Pythagorean identities come from the Pythagorean theorem applied to the unit circle:

• $\sin^2\theta + \cos^2\theta = 1$
• $1 + \tan^2\theta = \sec^2\theta$
• $1 + \cot^2\theta = \csc^2\theta$

The second and third are just rearrangements of the first. If you divide $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$, you get the tangent-secant version. Divide by $\sin^2\theta$ instead, and you get the cotangent-cosecant version.

Reciprocal identities pair each function with its reciprocal:

• $\csc\theta = \frac{1}{\sin\theta}$
• $\sec\theta = \frac{1}{\cos\theta}$
• $\cot\theta = \frac{1}{\tan\theta}$

Quotient identities express tangent and cotangent as ratios:

• $\tan\theta = \frac{\sin\theta}{\cos\theta}$
• $\cot\theta = \frac{\cos\theta}{\sin\theta}$

How to verify an identity: The key rule is that you work each side independently. You never move terms across the equals sign the way you would when solving an equation. Instead, you manipulate one or both sides until they match.

Here's a general approach:

1. Start with the more complicated side.
2. Look for opportunities to apply identities (especially converting everything to sine and cosine).
3. Use algebraic techniques like factoring, combining fractions, or multiplying by a conjugate.
4. Simplify until that side matches the other.

Example: Verify that $\tan\theta \cdot \cos\theta = \sin\theta$.

1. Start with the left side: $\tan\theta \cdot \cos\theta$
2. Replace $\tan\theta$ using the quotient identity: $\frac{\sin\theta}{\cos\theta} \cdot \cos\theta$
3. Cancel $\cos\theta$: $\sin\theta$
4. The left side now equals the right side. The identity is verified.

Simplifying Trigonometric Expressions

Simplification of complex trigonometric expressions

Simplifying trig expressions uses the same identities from above, combined with standard algebra moves. Here are the main techniques:

Algebraic techniques:

• Factor out common terms: $2\sin\theta + 4\cos\theta = 2(\sin\theta + 2\cos\theta)$
• Combine like terms: $3\sin^2\theta + 5\sin^2\theta = 8\sin^2\theta$
• Multiply by a strategic form of 1 (like $\frac{\sin\theta}{\sin\theta}$ or $\frac{1 - \cos\theta}{1 - \cos\theta}$) to create common denominators or useful products

Pythagorean substitutions are especially powerful. You can rearrange the Pythagorean identities to swap terms:

• $\sin^2\theta = 1 - \cos^2\theta$ and $\cos^2\theta = 1 - \sin^2\theta$
• $\tan^2\theta = \sec^2\theta - 1$
• $\cot^2\theta = \csc^2\theta - 1$

Rewriting in terms of sine and cosine often reveals simplifications that aren't obvious otherwise. Replace $\sec\theta$ with $\frac{1}{\cos\theta}$, $\csc\theta$ with $\frac{1}{\sin\theta}$, and so on.

Example: Simplify $\sec\theta - \cos\theta$.

1. Rewrite $\sec\theta$: $\frac{1}{\cos\theta} - \cos\theta$

2. Get a common denominator: $\frac{1 - \cos^2\theta}{\cos\theta}$

3. Apply the Pythagorean identity ($1 - \cos^2\theta = \sin^2\theta$): $\frac{\sin^2\theta}{\cos\theta}$

That's the simplified form. You could also write it as $\sin\theta \cdot \tan\theta$ if needed.

Patterns in trigonometric transformations

These additional formulas show up frequently in simplification problems. You don't always need them for basic identity work, but they're worth knowing.

Double-angle formulas:

• $\sin(2\theta) = 2\sin\theta\cos\theta$
• $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

The cosine double-angle formula has three equivalent forms. Pick whichever version creates the most useful substitution for your problem.

Half-angle formulas (derived from the double-angle formulas by solving for the squared half-angle):

• $\sin^2\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}$
• $\cos^2\!\left(\frac{\theta}{2}\right) = \frac{1 + \cos\theta}{2}$

Sum and difference formulas:

• $\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$
• $\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

Notice the sign pattern: for sine, the $\pm$ on the left matches the $\pm$ on the right. For cosine, the signs flip ($\mp$).

Fundamental Concepts in Trigonometry

The six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) all relate back to the unit circle, where any point on the circle has coordinates $(\cos\theta, \sin\theta)$. This is why $\sin^2\theta + \cos^2\theta = 1$ works: it's just the equation of a circle with radius 1.

Radian measure is the standard in most identity and equation work. If you see $\theta$ without a degree symbol, assume radians. One full revolution is $2\pi$ radians (equivalent to 360°).

When you move into solving trigonometric equations later in this unit, you'll combine these identities with inverse trig functions to find specific angle values. The simplification skills you build here make that process much more manageable.