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Chapter 5: Problem 43

Verify each identity. $$\tan \left(\theta+\frac{\pi}{4}\right)=\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}$$

Short Answer

Expert verified
Yes, the trigonometric identity is correct. You can show that \( \tan(\theta + \frac{\pi}{4}) = \frac{\cos\theta+\sin\theta}{\cos\theta-\sin\theta} \) by using the addition formula for tangent and simplifying.

Step by step solution

01

Add the Angle Formula for Tangent

The addition formula for tangent is \( \tan(A+B) = \frac{\sin A}{\cos B} + \frac{\cos A}{\sin B}\). Substitute \( \theta \) for \( A \) and \( \frac{\pi}{4} \) for \( B \) in the addition formula. So, \( \tan(\theta+\frac{\pi}{4})=\frac{\tan\theta+\tan\frac{\pi}{4}}{1-\tan\theta\tan\frac{\pi}{4}} \).
02

Substitute Tangent values

We know that \( \tan\frac{\pi}{4}=1 \). Substitute \( 1 \) into the equation from Step 1: \( =\frac{\tan\theta+1}{1-\tan\theta}\).
03

Convert Tangent to Sine/Cosine

Use the definition of tangent (\( \tan\theta=\frac{\sin\theta}{\cos\theta} \)). Substitute into the equation from Step 2: \( =\frac{\frac{\sin\theta}{\cos\theta}+1}{1-\frac{\sin\theta}{\cos\theta}} \)
04

Simplify

To simplify the equation, multiply the numerator and the denominator by \( \cos\theta \): \( =\frac{\sin\theta+\cos\theta}{\cos\theta-\sin\theta} \). This gives us the right side of the original equation, confirming the identity is correct.

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Most popular questions from this chapter

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\tan x=-4.7143$$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$2 \sin ^{2} x=3-\sin x$$

Describe the difference between verifying a trigonometric identity and solving a trigonometric equation.

Exercises \(116-118\) will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$\cos \frac{\pi}{2} \cos \frac{\pi}{3}=\frac{1}{2}\left[\cos \left(\frac{\pi}{2}-\frac{\pi}{3}\right)+\cos \left(\frac{\pi}{2}+\frac{\pi}{3}\right)\right]$$

Use a sketch to find the exact value of \(\sec \left(\sin ^{-1} \frac{1}{2}\right)\).
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