prove cot x + tan x / tan x - cot x = -sec2x

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.

---

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.

prove cot x + tan x / tan x - cot x = -sec^2x

Explore the step-by-step solution to proving the trigonometric identity: (cot x + tan x) / (tan x - cot x) = -sec^2 x. This problem covers important concepts in trigonometry, including trigonometric identities, simplification techniques, and key theorems like the Pythagorean identity and double angle identity. Suitable for high school students in Grades 10-12.

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