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Chapter 5: Problem 66

Prove the identity. $$\tan \left(\frac{\pi}{4}-\theta\right)=\frac{1-\tan \theta}{1+\tan \theta}$$

Short Answer

Expert verified
The given trigonometric identity has been successfully proven to be true based on algebraic manipulations.

Step by step solution

01

Write down the given identity

We start with the identity we need to prove \( \tan \left(\frac{\pi}{4}-\theta\right)=\frac{1-\tan \theta}{1+\tan \theta} \)
02

Apply the formula for tan(a - b)

The tangent of the difference of two angles can be expressed as \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\). Applying this formula results in, \( \tan \left(\frac{\pi}{4}-\theta\right) = \frac{\tan(\frac{\pi}{4})- \tan(\theta)}{1+ \tan(\frac{\pi}{4}) \tan( \theta) } \)
03

Evaluating tan(pi/4)

We know that \( \tan(\frac{\pi}{4}) = 1 \). Substituting this into the equation, we get \( \frac{1 - \tan( \theta)}{1 + 1 * \tan( \theta)} = \frac{1 - \tan \theta}{1 + \tan \theta} \)
04

Verify the equivalence

Now, if the expression on the left hand side is equal to the expression on the right hand side, then the identity is proven. Hence, the given identity is proven to be true.

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Most popular questions from this chapter

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