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Chapter 6: Problem 37

Verify the identity. $$\frac{\csc (-x)}{\sec (-x)}=-\cot x$$

Short Answer

Expert verified
Given identity \(\frac{\csc (-x)}{\sec (-x)}=-\cot x\) is true as both the sides of the equation have been proven to be equal.

Step by step solution

01

Express in terms of sine, cosine and cotangent

Rewrite \(\csc (-x)\) and \(\sec (-x)\) in terms of sine and cosine. By definition, \(\csc (-x) = \frac{1}{\sin(-x)}\) and \(\sec(-x) = \frac{1}{\cos(-x)}\). Thus the given identity becomes: \(\frac{1}{\sin(-x)}*\frac{\cos(-x)}{1}= -\cot x\)
02

Simplify Expression

The expression can be simplified to \(\frac{\cos(-x)}{\sin(-x)} = -\cot x\). Now we multiply both the sides by -1 to remove the negatives, the equation transforms to: \(-\frac{\cos(-x)}{\sin(-x)} = \cot x\)
03

Using Negative Angle Identities

Using the negative angle identities, which state that \(\cos(-x) = \cos x\) and \(\sin(-x) = -\sin x\), we can rewrite the left side as \(-\frac{\cos(x)}{-\sin(x)}\). This simplifies to \(\frac{\cos(x)}{\sin(x)}\).
04

Finalize the Transformation

The expression \(\frac{\cos(x)}{\sin(x)}\) is equivalent to \(\cot x\), which matches the right side of the original equation.

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