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Chapter 5: Problem 41

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\sin \phi(\csc \phi-\sin \phi)$$

Short Answer

Expert verified
The simplified expression is \( \sin \phi - \sin^2 \phi \).

Step by step solution

01

Understanding the given expression

The given expression is \( \sin \phi(\csc \phi-\sin \phi) \). Here, we notice that \(\csc \phi\) is just the reciprocal of \( \sin \phi \). So the expression can be rewritten in terms of sine only.
02

Rewrite the expression using identity

We know that \( \sin \phi \cdot \csc \phi = 1 \) because \( \csc \phi = 1/ \sin \phi \). Rewriting the expression, we get \( \sin \phi \cdot 1 - \sin \phi \cdot \sin \phi = \sin \phi - \sin^2 \phi \).
03

Simplify the expression

Here, we get the simplified result. Now there is no option to simplify it further.

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