Problem 46 - Precalculus Essentials

Question

Verify each identity. $$(\csc x-\cot x)^{2}=\frac{1-\cos x}{1+\cos x}$$

Step-by-Step Solution

Answer

Verified, the identity \((\csc x-\cot x)^{2}=\frac{1- \cos x}{1+ \cos x}\) is true.

Step 1: Expressing the left-hand side in terms of sine and cosine

The left-hand side of the equation is \((\csc x - \cot x)^{2}\). We first need to express \(\csc x\) and \(\cot x\) in terms of sine and cosine, since \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). Therefore, the left-hand side becomes \(\left(\frac{1}{\sin x} - \frac{\cos x}{\sin x}\right)^{2}\).

Step 2: Simplifying and squaring the equation

After expressing the left-hand side in terms of sine and cosine, it can be simplified and written as \(\left(\frac{1 - \cos x}{\sin x}\right)^{2}\). When squared, this equation becomes \(\frac{(1 - \cos x)^{2}}{\sin^{2} x}\) .

Step 3: Using the trigonometric identity

To simplify this equation further, use the identity \(\csc^{2} x = 1 + \cot^{2} x\), which gives us \(\sin^{2} x = 1 - \cos^{2} x\). Substitute \(\sin^{2} x\) in the denominator of our equation with \(1 - \cos^{2} x\). So, the equation becomes \(\frac{(1 - \cos x)^{2}}{1 - \cos^{2} x}\)

Step 4: Simplifying the equation to match the right-hand side

Now, simplifying this equation gives \(\frac{1 - \cos x}{1 + \cos x}\), which is the right-hand side. Hence, the identity is verified.