tan2(𝑥)−cot2
(𝑥)
csc2(𝑥) sec2(𝑥)
= −cos(2𝑥)

To simplify and verify the given expression:

\[
\frac{\tan^2(x) - \cot^2(x)}{\csc^2(x) \sec^2(x)} = -\cos(2x)
\]

let's break it down step-by-step.

1. **Rewrite the trigonometric functions in terms of sine and cosine:**

\[
\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)}, \quad \csc(x) = \frac{1}{\sin(x)}, \quad \sec(x) = \frac{1}{\cos(x)}
\]

2. **Square the functions:**

\[
\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}, \quad \cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)}, \quad \csc^2(x) = \frac{1}{\sin^2(x)}, \quad \sec^2(x) = \frac{1}{\cos^2(x)}
\]

3. **Rewrite the numerator:**

\[
\tan^2(x) - \cot^2(x) = \frac{\sin^2(x)}{\cos^2(x)} - \frac{\cos^2(x)}{\sin^2(x)}
\]

4. **Rewrite the denominator:**

\[
\csc^2(x) \sec^2(x) = \left(\frac{1}{\sin^2(x)}\right) \left(\frac{1}{\cos^2(x)}\right) = \frac{1}{\sin^2(x) \cos^2(x)}
\]

5. **Combine the numerator and the denominator:**

\[
\frac{\tan^2(x) - \cot^2(x)}{\csc^2(x) \sec^2(x)} = \frac{\frac{\sin^2(x)}{\cos^2(x)} - \frac{\cos^2(x)}{\sin^2(x)}}{\frac{1}{\sin^2(x) \cos^2(x)}} = \left(\sin^2(x) \cos^2(x)\right) \left(\frac{\sin^2(x)}{\cos^2(x)} - \frac{\cos^2(x)}{\sin^2(x)}\right)
\]

6. **Simplify the expression inside the parenthesis:**

\[
= \sin^2(x) \cos^2(x) \left(\frac{\sin^4(x) - \cos^4(x)}{\sin^2(x) \cos^2(x)}\right)
\]

Since \(\sin^2(x) \cos^2(x)\) in the denominator cancels with the one outside:

\[
= \sin^4(x) - \cos^4(x)
\]

7. **Factor the difference of squares:**

\[
\sin^4(x) - \cos^4(x) = (\sin^2(x) + \cos^2(x))(\sin^2(x) - \cos^2(x))
\]

Since \(\sin^2(x) + \cos^2(x) = 1\):

\[
= 1 \cdot (\sin^2(x) - \cos^2(x)) = \sin^2(x) - \cos^2(x)
\]

8. **Recall the double-angle identity:**

\[
\cos(2x) = \cos^2(x) - \sin^2(x)
\]

So, 

\[
\sin^2(x) - \cos^2(x) = -(\cos^2(x) - \sin^2(x)) = -\cos(2x)
\]

Hence:

\[
\frac{\tan^2(x) - \cot^2(x)}{\csc^2(x) \sec^2(x)} = -\cos(2x)
\]

Therefore, the given equation is verified:

\[
\frac{\tan^2(x) - \cot^2(x)}{\csc^2(x) \sec^2(x)} = -\cos(2x)
\]

Would you like further details or have any questions?

### Related Questions

1. What are some other identities involving \(\cos(2x)\)?
2. How can you derive \(\sin(2x)\) from the double-angle formulas?
3. What are the steps to prove \(\tan^2(x) + 1 = \sec^2(x)\)?
4. How do you simplify expressions involving \(\csc(x)\) and \(\sec(x)\)?
5. Can you provide an example of using trigonometric identities to simplify an integral?
6. What is the difference between \(\cot(x)\) and \(\tan(x)\)?
7. How do the Pythagorean identities relate to the double-angle formulas?
8. Can you show how to derive \(\cot^2(x) + 1 = \csc^2(x)\)?

### Tip
Always look for opportunities to use known identities to simplify trigonometric expressions. This can make complex problems much more manageable.

Learn how to simplify the trigonometric expression tan^2(x) - cot^2(x) / csc^2(x) sec^2(x) using trigonometric identities and the double-angle identity. Detailed step-by-step solution provided.

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