To verify if the given equation 

\[
\frac{1 + \tan^2 x}{1 + \cot^2 x} = \frac{(1 - \tan x)^2}{(1 - \cot x)^2}
\]

is true, let's simplify both sides and compare them.

### Simplifying the Left Side:

The left side is:

\[
\frac{1 + \tan^2 x}{1 + \cot^2 x}
\]

Using the identity \(\tan^2 x = \sec^2 x - 1\) and \(\cot^2 x = \csc^2 x - 1\), we can rewrite it as:

\[
\frac{\sec^2 x}{\csc^2 x} = \frac{\frac{1}{\cos^2 x}}{\frac{1}{\sin^2 x}} = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x
\]

### Simplifying the Right Side:

The right side is:

\[
\frac{(1 - \tan x)^2}{(1 - \cot x)^2}
\]

Using \(\cot x = \frac{1}{\tan x}\), the right side becomes:

\[
\frac{(1 - \tan x)^2}{\left(1 - \frac{1}{\tan x}\right)^2} = \frac{(1 - \tan x)^2}{\left(\frac{\tan x - 1}{\tan x}\right)^2} = \frac{(1 - \tan x)^2 \tan^2 x}{(\tan x - 1)^2} = \tan^2 x
\]

Thus, simplifying both sides gives us the same result: \(\tan^2 x\).

Therefore, the given equation is true:

\[
\frac{1 + \tan^2 x}{1 + \cot^2 x} = \frac{(1 - \tan x)^2}{(1 - \cot x)^2}
\]

Would you like to see more details or have any specific questions about this? Here are some related questions:

1. What are the identities involving tangent and cotangent?
2. How do you simplify complex trigonometric expressions?
3. Can we prove the given equation by another method?
4. What are other forms of verifying trigonometric identities?
5. How does \(\sec^2 x = 1 + \tan^2 x\) and \(\csc^2 x = 1 + \cot^2 x\) simplify equations?

**Tip:** Always use trigonometric identities to simplify and compare both sides of an equation for verification.

Learn how to verify the trigonometric identity (1 + tan^2)/(1 + cot^2) = (1 - tan)^2/(1 - cot)^2 by simplifying both sides. This problem covers key trigonometric concepts such as the Pythagorean identities for secant, cosecant, tangent, and cotangent. Suitable for students in Grades 10-12.

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