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

New Concept

This lesson introduces trigonometric identities—the fundamental rules of trigonometry. You'll learn to use these identities, like the Pythagorean and even-odd rules, to verify trigonometric statements and simplify complex expressions using algebraic strategies.

What’s next

Next up, you'll see these identities in action through interactive examples. Then, you'll master simplification and verification with a series of practice cards.

Property

The Pythagorean identities are equations involving trigonometric functions based on the properties of a right triangle. The second and third identities can be obtained by manipulating the first.

Examples

• To simplify $\csc^2 \theta - \cot^2 \theta$, we use the identity $1 + \cot^2 \theta = \csc^2 \theta$. Substituting gives $(1 + \cot^2 \theta) - \cot^2 \theta = 1$.
• To verify the identity $\frac{\sec^2 \theta - 1}{\sec^2 \theta} = \sin^2 \theta$, we can substitute $\tan^2 \theta$ for $\sec^2 \theta - 1$. This gives $\frac{\tan^2 \theta}{\sec^2 \theta} = \frac{\sin^2 \theta / \cos^2 \theta}{1 / \cos^2 \theta} = \sin^2 \theta$.
• To rewrite $\sin^2 \theta$ in terms of $\cos \theta$, we use $\sin^2 \theta + \cos^2 \theta = 1$. Rearranging gives $\sin^2 \theta = 1 - \cos^2 \theta$.

Explanation

These powerful identities come from the Pythagorean theorem ($a^2 + b^2 = c^2$) applied to the unit circle. They are essential for rewriting expressions, like converting between $\sin^2\theta$ and $\cos^2\theta$, to simplify or solve equations.

Property

The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle. An even function is one in which $f(-x) = f(x)$, and an odd function is one in which $f(-x) = -f(x)$.

Examples

• To simplify $\tan(-x) \cos(-x)$, we use the identities to get $(-\tan x)(\cos x) = -\frac{\sin x}{\cos x}(\cos x) = -\sin x$.
• To verify $(1 + \cos x)[1 + \cos(-x)] = (1+\cos x)^2$, we work on the left side: $(1 + \cos x)(1 + \cos x) = (1 + \cos x)^2$.
• To evaluate $\sin(-\frac{\pi}{6})$, we use the odd identity for sine: $\sin(-\frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2}$.

Explanation

Remember that cosine and its reciprocal, secant, are the only even functions; they 'absorb' the negative sign. The other four functions are odd and pass the negative sign out front. This is useful for simplifying expressions with negative angles.

Property

The reciprocal identities relate trigonometric functions that are reciprocals of each other. They are derived from the definitions of the basic trigonometric functions.

Examples

• To simplify $\tan x \cot x$, we substitute $\frac{1}{\tan x}$ for $\cot x$, which gives $\tan x (\frac{1}{\tan x}) = 1$.
• To rewrite $\frac{1}{\cos \theta}$ as a single function, we use the reciprocal identity to get $\sec \theta$.
• To simplify $\frac{\sec \alpha}{\csc \alpha}$, we can write it as $\frac{1/\cos \alpha}{1/\sin \alpha} = \frac{1}{\cos \alpha} \cdot \frac{\sin \alpha}{1} = \tan \alpha$.

Explanation

Think of these as function pairs that flip upside down. Sine and cosecant are a pair, cosine and secant are a pair, and tangent and cotangent are a pair. This allows you to switch between them to simplify expressions.

Property

The quotient identities define relationships among certain trigonometric functions and can be very helpful in verifying other identities. They are derived from the definitions of the basic trigonometric functions.

Examples

• To verify $\cot \theta \sin \theta = \cos \theta$, we substitute for $\cot \theta$: $(\frac{\cos \theta}{\sin \theta}) \sin \theta = \cos \theta$.
• To simplify $\tan x \csc x$, we rewrite everything in terms of sine and cosine: $(\frac{\sin x}{\cos x}) (\frac{1}{\sin x}) = \frac{1}{\cos x} = \sec x$.
• To rewrite $\cot \theta + \tan \theta$ as a single fraction: $\frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}$.

Explanation

These identities are your go-to move for breaking down tangent and cotangent into the fundamental building blocks of sine and cosine. This is often the first step in simplifying a complex expression or proving another identity.

Property

To verify an identity, transform one side of the equation until it is the same as the other side.

1. Start with the more complex side.
2. Look for chances to factor, add fractions, or square binomials.
3. Use known identities to make substitutions.
4. If stuck, try converting all terms to sines and cosines.

Examples

• Verify $(\sin x + \cos x)^2 = 1 + 2\sin x \cos x$. Expanding the left side gives $\sin^2 x + 2\sin x \cos x + \cos^2 x$. Since $\sin^2 x + \cos^2 x = 1$, this simplifies to $1 + 2\sin x \cos x$.
• Verify $\cos x (\tan x - \sec(-x)) = \sin x - 1$. The left side is $\cos x (\frac{\sin x}{\cos x} - \sec x) = \cos x (\frac{\sin x - 1}{\cos x}) = \sin x - 1$.
• Verify $\cos x - \cos^3 x = \cos x \sin^2 x$. Factoring the left side gives $\cos x (1 - \cos^2 x)$. Using the Pythagorean identity, this becomes $\cos x (\sin^2 x)$.

Explanation

Verifying an identity is like solving a puzzle. You are not solving for a variable, but proving two expressions are always equal. Use algebra and other identities to transform one side until it perfectly matches the other side.

Property

Algebraic patterns can simplify trigonometric expressions.
Difference of Squares: An expression like $a^2 - b^2 = (a-b)(a+b)$ can apply to expressions such as $\sin^2 x - 1$ or $4\sec^2\theta - 9$.
Quadratic Form: An expression like $2\cos^2 \theta + \cos \theta - 1$ is in the form $2x^2 + x - 1$ where $x = \cos \theta$, and can be factored.

Examples

• Rewrite $9\cos^2 \theta - 1$ using the difference of squares. This is $(3\cos \theta)^2 - 1^2$, which factors to $(3\cos \theta - 1)(3\cos \theta + 1)$.
• Factor $2\sin^2 \theta + \sin \theta - 1$. Letting $x = \sin \theta$, the expression becomes $2x^2 + x - 1$, which factors to $(2x - 1)(x + 1)$. So, the original expression is $(2\sin \theta - 1)(\sin \theta + 1)$.
• Simplify $(\sec x + 1)(\sec x - 1)$. This follows the difference of squares pattern, resulting in $\sec^2 x - 1$, which simplifies to $\tan^2 x$ by the Pythagorean identity.

Explanation

Don't forget your algebra skills! Recognizing familiar patterns like the difference of squares or quadratic equations is a secret weapon. It makes complex-looking trigonometric problems much easier to factor, simplify, and solve.