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Chapter 5: Problem 45

Verify the identity. $$\sin t \csc \left(\frac{\pi}{2}-t\right)=\tan t$$

Short Answer

Expert verified
\(\sin t \csc \left(\frac{\pi}{2}-t\right)=\tan t\) is a valid identity.

Step by step solution

01

Write down the given identity

We start with the given equation: \(\sin t \csc \left(\frac{\pi}{2}-t\right) = \tan t\)
02

Substitute the co-function identity \(\csc(\frac{\pi}{2} - t) = \sec t\)

By using the co-function identity \(\csc(\frac{\pi}{2} - t) = \sec t\), the equation becomes: \(\sin t \sec t\)
03

Convert secant to its equivalent

The secant function can be rewritten in terms of cosine as \( \sec t = \frac{1}{\cos t} \). Therefore the equation becomes: \(\sin t \cdot \frac{1}{\cos t} = \frac{\sin t}{\cos t} \)
04

Simplification

The equation \( \frac{\sin t}{\cos t}\) simplifies to \( \tan t \), which is the right side of our original equation, hence verified.

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