<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Simplifying Trigonometric Expressions

Convert to sine/cosine and use basic trig identities to simplify.

Atoms Practice
Estimated11 minsto complete
%
Progress
Practice Simplifying Trigonometric Expressions
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated11 minsto complete
%
Practice Now
Turn In
Simplifying Trigonometric Expressions

How could you write the trigonometric function \begin{align*}\cos\theta + \cos\theta(\tan^2\theta)\end{align*}cosθ+cosθ(tan2θ) more simply?

Simplifying Trigonometric Expressions

Now that you are more familiar with trig identities, we can use them to simplify expressions. Remember, that you can use any of the following identities.

Reciprocal Identities: \begin{align*}\csc \theta=\frac{1}{\sin \theta}, \sec \theta=\frac{1}{\cos \theta},\end{align*}cscθ=1sinθ,secθ=1cosθ, and \begin{align*}\cot \theta=\frac{1}{\tan \theta}\end{align*}cotθ=1tanθ

Tangent and Cotangent Identities: \begin{align*}\tan \theta=\frac{\sin \theta}{\cos \theta}\end{align*}tanθ=sinθcosθ and \begin{align*}\cot \theta=\frac{\cos \theta}{\sin \theta}\end{align*}cotθ=cosθsinθ

Pythagorean Identities: \begin{align*}\sin^2 \theta+ \cos^2 \theta=1, 1+ \tan^2 \theta=\sec^2 \theta,\end{align*}sin2θ+cos2θ=1,1+tan2θ=sec2θ, and \begin{align*}1+ \cot^2 \theta=\csc^2 \theta\end{align*}1+cot2θ=csc2θ

Cofunction Identities: \begin{align*}\sin \left(\frac{\pi}{2} - \theta\right)=\cos \theta, \cos \left(\frac{\pi}{2} - \theta\right)=\sin \theta,\end{align*}sin(π2θ)=cosθ,cos(π2θ)=sinθ, and \begin{align*}\tan \left(\frac{\pi}{2} - \theta\right)=\cot \theta\end{align*}tan(π2θ)=cotθ

Negative Angle Identities: \begin{align*}\sin(- \theta)=- \sin \theta, \cos(- \theta)=\cos \theta,\end{align*}sin(θ)=sinθ,cos(θ)=cosθ, and \begin{align*}\tan(- \theta)=- \tan \theta\end{align*}tan(θ)=tanθ

Let's simplify the following expressions.

  1. \begin{align*}\frac{\sec x}{\sec x-1}\end{align*}secxsecx1

When simplifying trigonometric expressions, one approach is to change everything into sine or cosine. First, we can change secant to cosine using the Reciprocal Identity.

\begin{align*}\frac{\sec x}{\sec x - 1} \rightarrow \frac{\frac{1}{\cos x}}{\frac{1}{\cos x}-1}\end{align*}secxsecx11cosx1cosx1

Now, combine the denominator into one fraction by multiplying 1 by \begin{align*}\frac{\cos x}{\cos x}\end{align*}cosxcosx.

\begin{align*}\frac{\frac{1}{\cos x}}{\frac{1}{\cos x}-1} \rightarrow \frac{\frac{1}{\cos x}}{\frac{1}{\cos x}- \frac{\cos x}{\cos x}} \rightarrow \frac{\frac{1}{\cos x}}{\frac{1- \cos x}{\cos x}}\end{align*}1cosx1cosx11cosx1cosxcosxcosx1cosx1cosxcosx

Change this problem into a division problem and simplify.

\begin{align*}\frac{\frac{1}{\cos x}}{\frac{1-\cos x}{\cos x}} \rightarrow & \frac{1}{\cos x} \div \frac{1- \cos x}{\cos x} \\ & \frac{1}{\cancel{\cos x}} \cdot \frac{\cancel{\cos x}}{1- \cos x} \\ & \frac{1}{1- \cos x}\end{align*}1cosx1cosxcosx1cosx÷1cosxcosx1cosxcosx1cosx11cosx

  1. \begin{align*}\frac{\sin^4x- \cos^4x}{\sin^2x- \cos^2x}\end{align*}sin4xcos4xsin2xcos2x

With this problem, we need to factor the numerator and denominator and see if anything cancels. The rules of factoring a quadratic and the special quadratic formulas can be used in this scenario.

\begin{align*}\frac{\sin^4x - \cos^4x}{\sin^2x-\cos^2x} \rightarrow \frac{\cancel{\left(\sin^2x-\cos^2x\right)} \left(\sin^2x+ \cos^2x\right)}{\cancel{\left(\sin^2x-\cos^2x\right)}} \rightarrow \sin^2x+\cos^2x \rightarrow 1\end{align*}sin4xcos4xsin2xcos2x(sin2xcos2x)(sin2x+cos2x)(sin2xcos2x)sin2x+cos2x1

In the last step, we simplified to the left hand side of the Pythagorean Identity. Therefore, this expression simplifies to 1.

  1. \begin{align*}\sec \theta \tan^2 \theta+\sec \theta\end{align*}

First, pull out the GCF.

\begin{align*}\sec \theta \tan^2 \theta+ \sec \theta \rightarrow \sec \theta(\tan^2 \theta+1)\end{align*}

Now, \begin{align*}\tan^2 \theta+1=\sec^2 \theta\end{align*} from the Pythagorean Identities, so simplify further.

\begin{align*}\sec \theta(\tan^2 \theta+1) \rightarrow \sec \theta \cdot \sec^2 \theta \rightarrow \sec^3 \theta\end{align*}

Examples

Example 1

Earlier, you were asked to simplify the trigonometric function \begin{align*}\cos\theta + \cos\theta(\tan^2\theta)\end{align*}.

Notice that the terms in the expression \begin{align*}\cos\theta + \cos\theta(\tan^2\theta)\end{align*} have a common factor of \begin{align*}\cos\theta\end{align*}, so start by factoring this common term out.

\begin{align*}\cos\theta + \cos\theta(\tan^2\theta)\\ \cos\theta(1 + tan^2\theta)\end{align*}

Now, use the trigonometric identity \begin{align*}1+ \tan^2 \theta=\sec^2 \theta\end{align*}, substitute, and simplify.

\begin{align*}\cos\theta(1 + tan^2\theta)\\ =\cos\theta(sec^2\theta)\\ =\cos\theta(\frac{1}{cos^2\theta)}\\ =\frac{1}{cos\theta}\\ =\sec\theta\end{align*}

Simplify the following trigonometric expressions.

Example 2

\begin{align*}\cos \left(\frac{\pi}{2} - x\right) \cot x\end{align*}

Use the Cotangent Identity and the Cofunction Identity \begin{align*}\cos \left(\frac{\pi}{2}- \theta \right)=\sin \theta\end{align*}.

\begin{align*}\cos \left(\frac{\pi}{2}-x\right) \cot x \rightarrow \cancel{\sin x} \cdot \frac{\cos x}{\cancel{\sin x}} \rightarrow \cos x\end{align*}

Example 3

\begin{align*}\frac{\sin \left(-x\right) \cos x}{\tan x}\end{align*}

Use the Negative Angle Identity and the Tangent Identity.

\begin{align*}\frac{\sin \left(-x\right) \cos x}{\tan x} \rightarrow \frac{- \sin x \cos x}{\frac{\sin x}{\cos x}} \rightarrow - \cancel{\sin x} \cos x \cdot \frac{\cos x}{\sin x} \rightarrow - \cos^2x\end{align*}

Example 4

\begin{align*}\frac{\cot x \cos x}{\tan \left(-x\right) \sin \left(\frac{\pi}{2}-x \right)}\end{align*}

In this problem, you will use several identities.

\begin{align*}\frac{\cot x \cos x}{\tan \left(-x\right) \sin \left(\frac{\pi}{2}-x\right)} \rightarrow \frac{\frac{\cos x}{\sin x} \cdot \cos x}{- \frac{\sin x}{\cancel{\cos x}} \cdot \cancel{\cos x}} \rightarrow \frac{\frac{\cos^2 x}{\sin x}}{- \sin x} \rightarrow \frac{\cos^2x}{\sin x} \cdot - \frac{1}{\sin x} \rightarrow - \frac{\cos^2x}{\sin^2 x} \rightarrow - \cot^2x\end{align*}

Review

Simplify the following expressions.

  1. \begin{align*}\cot x \sin x\end{align*}
  2. \begin{align*}\cos^2x \tan(-x)\end{align*}
  3. \begin{align*}\frac{\cos \left(-x\right)}{\sin \left(-x\right)}\end{align*}
  4. \begin{align*}\sec x \cos(-x)- \sin^2x\end{align*}
  5. \begin{align*}\sin x(1+ \cot^2x)\end{align*}
  6. \begin{align*}1- \sin^2 \left(\frac{\pi}{2} - x\right)\end{align*}
  7. \begin{align*}1- \cos^2 \left(\frac{\pi}{2}-x\right)\end{align*}
  8. \begin{align*}\frac{\tan \left(\frac{\pi}{2}-x\right) \sec x}{1- \csc^2 x}\end{align*}
  9. \begin{align*}\frac{\cos^2x \tan^2x-1}{\cos^2x}\end{align*}
  10. \begin{align*}\cot^2x+ \sin^2x+ \cos^2(-x)\end{align*}
  11. \begin{align*}\frac{\sec x \sin x+ \cos \left(\frac{\pi}{2}-x\right)}{1+ \cos x}\end{align*}
  12. \begin{align*}\frac{\cos \left(-x\right)}{1+ \sin \left(-x\right)}\end{align*}
  13. \begin{align*}\frac{\sin^2 \left(-x\right)}{\tan^2x}\end{align*}
  14. \begin{align*}\tan \left(\frac{\pi}{2}-x\right) \cot x- \csc^2 x\end{align*}
  15. \begin{align*}\frac{\csc x \left(1- \cos^2x \right)}{\sin x \cos x}\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.8. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Simplifying Trigonometric Expressions.
Please wait...
Please wait...