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To prove the given trigonometric identity, let's analyze it step-by-step.

### Given equation:
\[
\sin^2 x (1 + \cot x) + \cos^2 x (1 - \tan x) + \cot^2 x = \csc^2 x
\]

We aim to prove that the left-hand side (LHS) equals the right-hand side (RHS). I'll go step by step, simplifying the terms.

### Step 1: Rewrite in terms of \(\sin x\) and \(\cos x\)
Start by rewriting \(\cot x\) and \(\tan x\) in terms of \(\sin x\) and \(\cos x\):

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

### Step 2: Expand and simplify each term
Let's break down each part of the expression.

#### First term:
\[
\sin^2 x (1 + \cot x) = \sin^2 x \left(1 + \frac{\cos x}{\sin x}\right) = \sin^2 x \left(\frac{\sin x + \cos x}{\sin x}\right) = \sin x (\sin x + \cos x)
\]

#### Second term:
\[
\cos^2 x (1 - \tan x) = \cos^2 x \left(1 - \frac{\sin x}{\cos x}\right) = \cos^2 x \left(\frac{\cos x - \sin x}{\cos x}\right) = \cos x (\cos x - \sin x)
\]

#### Third term:
\[
\cot^2 x = \frac{\cos^2 x}{\sin^2 x}
\]

### Step 3: Combine terms
Now, let's combine the results:
\[
\sin x (\sin x + \cos x) + \cos x (\cos x - \sin x) + \frac{\cos^2 x}{\sin^2 x}
\]

Simplifying the first two terms:
\[
= \sin^2 x + \sin x \cos x + \cos^2 x - \sin x \cos x + \frac{\cos^2 x}{\sin^2 x}
\]

The \(\sin x \cos x\) terms cancel out:
\[
= \sin^2 x + \cos^2 x + \frac{\cos^2 x}{\sin^2 x}
\]

Since \(\sin^2 x + \cos^2 x = 1\) (Pythagorean identity):
\[
= 1 + \frac{\cos^2 x}{\sin^2 x}
\]

### Step 4: Express in terms of \(\csc^2 x\)
The last term \(\frac{\cos^2 x}{\sin^2 x}\) is \(\cot^2 x\), so:
\[
1 + \cot^2 x
\]

But we know from the identity:
\[
1 + \cot^2 x = \csc^2 x
\]

### Step 5: Conclusion
Thus, the left-hand side simplifies to \(\csc^2 x\), which matches the right-hand side.

\[
\boxed{\csc^2 x = \csc^2 x}
\]

The given identity is proved.

---

Do you have any questions, or would you like more details? 

Here are five related questions to expand your understanding:
1. How can you prove the identity \(1 + \cot^2 x = \csc^2 x\)?
2. What are other common Pythagorean identities in trigonometry?
3. How do you simplify expressions involving \(\tan x\) and \(\cot x\)?
4. Can you derive \(\cot x\) and \(\csc x\) from basic trigonometric definitions?
5. What are the applications of trigonometric identities in calculus?

**Tip:** Always try to use fundamental trigonometric identities like \(\sin^2 x + \cos^2 x = 1\) early on when simplifying expressions.

Prove the trigonometric identity: sin^2(x)(1 + cot(x)) + cos^2(x)(1 - tan(x)) + cot^2(x) = csc^2(x).

Explore the proof of the trigonometric identity sin^2(x)(1 + cot(x)) + cos^2(x)(1 - tan(x)) + cot^2(x) = csc^2(x). Learn the key steps and fundamental trigonometric identities, including Pythagorean identities and simplifications of cot and tan functions.

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