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Chapter 5: Problem 33

Verify the identity. $$\tan \left(\frac{\pi}{2}-\theta\right) \tan \theta=1$$

Short Answer

Expert verified
The given trigonometric identity has been verified successfully.

Step by step solution

01

Use Cotangent Identity

Rewrite the expression \(\tan \left(\frac{\pi}{2}-\theta\right)\) using the identity \(\tan(\frac{\pi}{2} - x) = \cot(x)\). \n So, \(\tan \left(\frac{\pi}{2}-\theta\right) = \cot(\theta)\)
02

Use Cotangent Definition

Recall that \(\cot(\theta)\) can be written as \(\frac{1}{\tan(\theta)}\), from the definition of cotangent as the reciprocal of tangent. So, \(\cot(\theta) = \frac{1}{\tan(\theta)}\)
03

Substitute and Verify Identity

Substitute the value from step 2 into the identity. This gives: \(\cot(\theta) \cdot \tan(\theta) = \frac{1}{\tan(\theta)} \cdot \tan(\theta) = 1\). The result confirms that the given identity is correct.

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