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Simplifying Trigonometric Expressions

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Simplifying Trigonometric Expressions

How could you write the trigonometric function \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: \csc \theta=\frac{1}{\sin \theta}, \sec \theta=\frac{1}{\cos \theta}, and \cot \theta=\frac{1}{\tan \theta}

Tangent and Cotangent Identities: \tan \theta=\frac{\sin \theta}{\cos \theta} and \cot \theta=\frac{\cos \theta}{\sin \theta}

Pythagorean Identities: \sin^2 \theta+ \cos^2 \theta=1, 1+ \tan^2 \theta=\sec^2 \theta, and 1+ \cot^2 \theta=\csc^2 \theta

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

Negative Angle Identities: \sin(- \theta)=- \sin \theta, \cos(- \theta)=\cos \theta, and \tan(- \theta)=- \tan \theta

Example A

Simplify \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.

\frac{\sec x}{\sec x - 1} \rightarrow \frac{\frac{1}{\cos x}}{\frac{1}{\cos x}-1}

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

\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}}

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 \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.

\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

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

Example C

Simplify \sec \theta \tan^2 \theta+\sec \theta .

Solution: First, pull out the GCF.

\sec \theta \tan^2 \theta+ \sec \theta \rightarrow \sec \theta(\tan^2 \theta+1)

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

\sec \theta(\tan^2 \theta+1) \rightarrow \sec \theta \cdot \sec^2 \theta \rightarrow \sec^3 \theta

Concept Problem Revisit

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

\cos\theta + \cos\theta(\tan^2\theta)\\\cos\theta(1 + tan^2\theta)

Now, use the trigonmetric identity 1+ \tan^2 \theta=\sec^2 \theta , substitute, and simplify.

\cos\theta(1 + tan^2\theta)\\=\cos\theta(sec^2\theta)\\=\cos\theta(\frac{1}{cos^2\theta)}\\=\frac{1}{cos\theta}\\=\sec\theta

Guided Practice

Simplify the following trigonometric expressions.

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

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

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

Answers

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

\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.

\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.

\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

Practice

Simplify the following expressions.

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

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