Problem 66 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(\frac{\cos \frac{x}{2} }{\sin \frac{x}{2}}\).

Step 2: 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.

Step 3: 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.