We are given the equation:

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

Let's break it down and prove the identity step by step.

Left-hand side:

$(\csc x - \cot x)^2$

First, recall the trigonometric identities for $\csc x$ and $\cot x$:

• $\csc x = \frac{1}{\sin x}$
• $\cot x = \frac{\cos x}{\sin x}$

Now, expand the left-hand side:

$(\csc x - \cot x)^2 = \left(\frac{1}{\sin x} - \frac{\cos x}{\sin x}\right)^2 = \left(\frac{1 - \cos x}{\sin x}\right)^2$

This simplifies to:

$\frac{(1 - \cos x)^2}{\sin^2 x}$

Right-hand side:

$\frac{1 - \cos x}{1 + \cos x}$

The goal is to prove that both sides of the equation are equal, i.e., that:

$\frac{(1 - \cos x)^2}{\sin^2 x} = \frac{1 - \cos x}{1 + \cos x}$

Step-by-step verification:

1. Start with the right-hand side and multiply both the numerator and denominator by $1 - \cos x$ (this is a standard technique to simplify such expressions):

$\frac{1 - \cos x}{1 + \cos x} \times \frac{1 - \cos x}{1 - \cos x} = \frac{(1 - \cos x)^2}{(1 + \cos x)(1 - \cos x)}$

2. Use the difference of squares identity for the denominator:

$(1 + \cos x)(1 - \cos x) = 1^2 - (\cos x)^2 = 1 - \cos^2 x$

3. Now, the expression becomes:

$\frac{(1 - \cos x)^2}{1 - \cos^2 x}$

4. Recall the Pythagorean identity:

$1 - \cos^2 x = \sin^2 x$

Thus, the right-hand side becomes:

$\frac{(1 - \cos x)^2}{\sin^2 x}$

Conclusion:

Both sides are now equal:

$\frac{(1 - \cos x)^2}{\sin^2 x} = \frac{(1 - \cos x)^2}{\sin^2 x}$

Thus, the identity is proven.

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Would you like further clarification on any part of the solution? Or do you have additional questions? Here's a list of related questions you may find useful:

1. How can we derive other trigonometric identities using similar methods?
2. What are some common simplifications when working with trigonometric functions?
3. How can we prove identities involving secant and tangent?
4. Can we extend this method to more complex trigonometric expressions?
5. What is the significance of the Pythagorean identity in trigonometry?

Tip: When simplifying trigonometric identities, always look for common trigonometric identities and try to factor expressions where possible.