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# Simplifying Trigonometric Expressions

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

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Practice Simplifying Trigonometric Expressions
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Simplifying Trigonometric Expressions

How could you write the trigonometric function cosθ+cosθ(tan2θ)$\cos\theta + \cos\theta(\tan^2\theta)$ more simply?

### Guidance

Now that you are more familiar with trig identities, we can use them to simplify expressions. Remember, that you can use any of the identities in the Introduction to Trig Identities concept. Here is a list of the identities again:

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

Tangent and Cotangent Identities: tanθ=sinθcosθ$\tan \theta=\frac{\sin \theta}{\cos \theta}$ and cotθ=cosθsinθ$\cot \theta=\frac{\cos \theta}{\sin \theta}$

Pythagorean Identities: sin2θ+cos2θ=1,1+tan2θ=sec2θ,$\sin^2 \theta+ \cos^2 \theta=1, 1+ \tan^2 \theta=\sec^2 \theta,$ and 1+cot2θ=csc2θ$1+ \cot^2 \theta=\csc^2 \theta$

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

Negative Angle Identities: sin(θ)=sinθ,cos(θ)=cosθ,$\sin(- \theta)=- \sin \theta, \cos(- \theta)=\cos \theta,$ and tan(θ)=tanθ$\tan(- \theta)=- \tan \theta$

#### Example A

Simplify secxsecx1$\frac{\sec x}{\sec x-1}$ .

Solution: 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.

secxsecx11cosx1cosx1

Now, combine the denominator into one fraction by multiplying 1 by cosxcosx$\frac{\cos x}{\cos x}$ .

1cosx1cosx11cosx1cosxcosxcosx1cosx1cosxcosx

Change this problem into a division problem and simplify.

\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}

#### Example B

Simplify sin4xcos4xsin2xcos2x$\frac{\sin^4x- \cos^4x}{\sin^2x- \cos^2x}$ .

Solution: 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.

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.

#### Example C

Simplify secθtan2θ+secθ$\sec \theta \tan^2 \theta+\sec \theta$ .

Solution: First, pull out the GCF.

secθtan2θ+secθsecθ(tan2θ+1)

Now, tan2θ+1=sec2θ$\tan^2 \theta+1=\sec^2 \theta$ from the Pythagorean Identities, so simplify further.

secθ(tan2θ+1)secθsec2θsec3θ

Concept Problem Revisit

Notice that the terms in the expression cosθ+cosθ(tan2θ)$\cos\theta + \cos\theta(\tan^2\theta)$ have a common factor of cosθ$\cos\theta$ , so start by factoring this common term out.

cosθ+cosθ(tan2θ)cosθ(1+tan2θ)

Now, use the trigonometric identity 1+tan2θ=sec2θ$1+ \tan^2 \theta=\sec^2 \theta$ , substitute, and simplify.

cosθ(1+tan2θ)=cosθ(sec2θ)=cosθ(1cos2θ)=1cosθ=secθ

### Guided Practice

Simplify the following trigonometric expressions.

1. cos(π2x)cotx$\cos \left(\frac{\pi}{2} - x\right) \cot x$

2. sin(x)cosxtanx$\frac{\sin \left(-x\right) \cos x}{\tan x}$

3. cotxcosxtan(x)sin(π2x)$\frac{\cot x \cos x}{\tan \left(-x\right) \sin \left(\frac{\pi}{2}-x \right)}$

1. Use the Cotangent Identity and the Cofunction Identity cos(π2θ)=sinθ$\cos \left(\frac{\pi}{2}- \theta \right)=\sin \theta$ .

cos(π2x)cotxsinxcosxsinxcosx$\cos \left(\frac{\pi}{2}-x\right) \cot x \rightarrow \cancel{\sin x} \cdot \frac{\cos x}{\cancel{\sin x}} \rightarrow \cos x$

2. Use the Negative Angle Identity and the Tangent Identity.

sin(x)cosxtanxsinxcosxsinxcosxsinxcosxcosxsinxcos2x$\frac{\sin \left(-x\right) \cos x}{\tan x} \rightarrow \frac{- \sin x \cos x}{\frac{\sin x}{\cos x}} \rightarrow - \cancel{\sin x} \cos x \cdot \frac{\cos x}{\sin x} \rightarrow - \cos^2x$

3. In this problem, you will use several identities.

cotxcosxtan(x)sin(π2x)cosxsinxcosxsinxcosxcosxcos2xsinxsinxcos2xsinx1sinxcos2xsin2xcot2x$\frac{\cot x \cos x}{\tan \left(-x\right) \sin \left(\frac{\pi}{2}-x\right)} \rightarrow \frac{\frac{\cos x}{\sin x} \cdot \cos x}{- \frac{\sin x}{\cancel{\cos x}} \cdot \cancel{\cos x}} \rightarrow \frac{\frac{\cos^2 x}{\sin x}}{- \sin x} \rightarrow \frac{\cos^2x}{\sin x} \cdot - \frac{1}{\sin x} \rightarrow - \frac{\cos^2x}{\sin^2 x} \rightarrow - \cot^2x$

### Explore More

Simplify the following expressions.

1. cotxsinx$\cot x \sin x$
2. cos2xtan(x)$\cos^2x \tan(-x)$
3. cos(x)sin(x)$\frac{\cos \left(-x\right)}{\sin \left(-x\right)}$
4. secxcos(x)sin2x$\sec x \cos(-x)- \sin^2x$
5. sinx(1+cot2x)$\sin x(1+ \cot^2x)$
6. 1sin2(π2x)$1- \sin^2 \left(\frac{\pi}{2} - x\right)$
7. 1cos2(π2x)$1- \cos^2 \left(\frac{\pi}{2}-x\right)$
8. tan(π2x)secx1csc2x$\frac{\tan \left(\frac{\pi}{2}-x\right) \sec x}{1- \csc^2 x}$
9. cos2xtan2x1cos2x$\frac{\cos^2x \tan^2x-1}{\cos^2x}$
10. cot2x+sin2x+cos2(x)$\cot^2x+ \sin^2x+ \cos^2(-x)$
11. secxsinx+cos(π2x)1+cosx$\frac{\sec x \sin x+ \cos \left(\frac{\pi}{2}-x\right)}{1+ \cos x}$
12. cos(x)1+sin(x)$\frac{\cos \left(-x\right)}{1+ \sin \left(-x\right)}$
13. sin2(x)tan2x$\frac{\sin^2 \left(-x\right)}{\tan^2x}$
14. tan(π2x)cotxcsc2x$\tan \left(\frac{\pi}{2}-x\right) \cot x- \csc^2 x$
15. cscx(1cos2x)sinxcosx$\frac{\csc x \left(1- \cos^2x \right)}{\sin x \cos x}$