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Chapter 5: Problem 66

In Exercises \(59-68\), verify each identity. $$\cot \frac{x}{2}=\frac{1+\cos x}{\sin x}$$

Short Answer

Expert verified
Verified. The identity \(\cot \frac{x}{2}=\frac{1+\cos x}{\sin x}\) holds true.

Step by step solution

01

Express cotangent in terms of other trigonometric ratios

We know that \(\cot y = \frac{\cos y}{\sin y}\). Let's first write down the identity given by substituting \(\cot \frac{x}{2}\) with \(\frac{\cos \frac{x}{2} }{\sin \frac{x}{2}}\).
02

Use the half-angle identities

Half-angle identities are commonly used in trigonometric problems. They are as follows: \(\cos \frac{x}{2} = \sqrt{\frac{1+\cos x}{2}}\) and \(\sin \frac{x}{2} = \sqrt{\frac{1-\cos x}{2}}\). Let's substitute these into our expression.
03

Simplify the equation

After substitutions in step 2, we end up with \(\frac{\sqrt{\frac{1+\cos x}{2}}}{\sqrt{\frac{1-\cos x}{2}}}\). This simplifies to \(\sqrt{\frac{1+\cos x}{1-\cos x}}\). In order to simplify this further, we might want to rationalize the expression by multiplying the numerator and denominator by \(\sqrt{1+\cos x}\). This gives us \(\frac{1+\cos x}{\sin x}\) that is equal to the right hand side of the original identity. Hence, it is verified.

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