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Chapter 6: Problem 34

Find a formula for \(\tan \left(\theta-\frac{\pi}{4}\right)\).

Short Answer

Expert verified
The formula for \(\tan \left(\theta-\frac{\pi}{4}\right)\) is: \(\tan(\theta - \frac{\pi}{4}) = \frac{\tan(\theta) - 1}{1 + \tan(\theta)}\)

Step by step solution

01

Identify the given values

In our formula, we have \(x = \theta\) and \(y = \frac{\pi}{4}\). We also know that \(\tan{\frac{\pi}{4}} = 1\).
02

Apply the tangent subtraction formula

Using the tangent subtraction formula, we have: $$\tan(\theta - \frac{\pi}{4}) = \frac{\tan(\theta) - \tan(\frac{\pi}{4})}{1 + \tan(\theta)\tan(\frac{\pi}{4})}$$
03

Substitute the known values into the formula

Substitute the known value of \(\tan\frac{\pi}{4} = 1\) into the formula: $$\tan(\theta - \frac{\pi}{4}) = \frac{\tan(\theta) - 1}{1 + \tan(\theta) \cdot 1}$$
04

Simplify the formula

Simplify the formula further: $$\tan(\theta - \frac{\pi}{4}) = \frac{\tan(\theta) - 1}{1 + \tan(\theta)}$$ Hence, the formula for \(\tan \left(\theta-\frac{\pi}{4}\right)\) is: $$\tan(\theta - \frac{\pi}{4}) = \frac{\tan(\theta) - 1}{1 + \tan(\theta)}$$

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