We are given the equation to prove:

$\frac{\cot x + \tan x}{\tan x - \cot x} = -\sec^2 x$

Step-by-step proof:

Step 1: Express $\cot x$ and $\tan x$ in terms of $\sin x$ and $\cos x$

Recall the trigonometric identities for $\cot x$ and $\tan x$: $\cot x = \frac{\cos x}{\sin x}, \quad \tan x = \frac{\sin x}{\cos x}$

Substitute these into the left-hand side of the equation: $\frac{\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}}$

Step 2: Simplify the numerator and denominator

To simplify, let's combine the terms in both the numerator and denominator by obtaining a common denominator.

For the numerator: $\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{\cos^2 x + \sin^2 x}{\sin x \cos x}$ Using the Pythagorean identity $\cos^2 x + \sin^2 x = 1$, this becomes: $\frac{1}{\sin x \cos x}$

For the denominator: $\frac{\sin x}{\cos x} - \frac{\cos x}{\sin x} = \frac{\sin^2 x - \cos^2 x}{\sin x \cos x}$

Thus, we now have: $\frac{\frac{1}{\sin x \cos x}}{\frac{\sin^2 x - \cos^2 x}{\sin x \cos x}}$

Step 3: Simplify the fraction

The common factor $\sin x \cos x$ cancels out from both the numerator and denominator, leaving: $\frac{1}{\sin^2 x - \cos^2 x}$

Recall that $\sin^2 x - \cos^2 x = -\cos 2x$ (from the double angle identity). Thus, the expression becomes: $\frac{1}{-\cos 2x} = -\frac{1}{\cos 2x}$

Step 4: Use the identity for $\sec 2x$

We know that $\sec 2x = \frac{1}{\cos 2x}$, so the expression simplifies to: $-\sec^2 x$

Thus, we have proven that: $\frac{\cot x + \tan x}{\tan x - \cot x} = -\sec^2 x$

Conclusion:

The given identity is proven.

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Do you want any details or have questions on specific steps? Here are 5 related questions to explore further:

1. Can you prove $\sec^2 x = 1 + \tan^2 x$ using a right triangle?
2. What are the key steps in proving other trigonometric identities?
3. How would you prove $\frac{1}{\tan x} + \tan x = \sec^2 x$?
4. What happens if you substitute different angle values like $x = 45^\circ$ in this identity?
5. Can this identity be used in solving integration problems?

Tip: Always start simplifying trigonometric identities by converting into $\sin x$ and $\cos x$ if you're unsure where to start.