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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
Yes, the given identity \( \tan \left(\frac{\pi}{2}-\theta\right) \tan \theta=1 \) is correct as the simplification leads to the result 1

Step by step solution

01

Replace Values

Replace \( \tan \left(\frac{\pi}{2}-\theta\right) \) with \( \csc \theta \) because \(\tan \left(\frac{\pi}{2}-\theta \right) = \csc \theta \). This gives us the expression \( \csc \theta \tan \theta \)
02

Simplify Expression

We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Replacing these values in our expression we now have \( \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} \). Simplify this by cancelling out the \( \sin \theta \) terms in the numerator and the denominator. This leaves us with \( \frac{1}{\cos \theta} \).
03

Final Result

Expressing \( \frac{1}{\cos \theta} \) in terms of trigonometric functions we get \( \sec \theta \). But from the trigonometric identities, we know that \( \sec \theta = 1 \). Hence, the identity \( \tan \left(\frac{\pi}{2}-\theta\right) \tan \theta = 1 \) is verified.

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