cot^2x\:sin^2x\:+\:sin^2\:x-\:cos^2\:x\:=\:sin^2\:x

We are tasked with simplifying or verifying the given equation:

\[
\cot^2 x \cdot \sin^2 x + \sin^2 x - \cos^2 x = \sin^2 x
\]

### Step 1: Express \(\cot^2 x\) in terms of \(\sin x\) and \(\cos x\)
We know the identity for cotangent:
\[
\cot x = \frac{\cos x}{\sin x}
\]
Thus,
\[
\cot^2 x = \frac{\cos^2 x}{\sin^2 x}
\]

### Step 2: Substitute \(\cot^2 x\) in the equation
Substitute \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\) into the given equation:

\[
\frac{\cos^2 x}{\sin^2 x} \cdot \sin^2 x + \sin^2 x - \cos^2 x = \sin^2 x
\]

### Step 3: Simplify
The term \(\frac{\cos^2 x}{\sin^2 x} \cdot \sin^2 x\) simplifies to:
\[
\cos^2 x
\]
So, the equation becomes:
\[
\cos^2 x + \sin^2 x - \cos^2 x = \sin^2 x
\]

### Step 4: Cancel out terms
The \(\cos^2 x\) terms cancel out:
\[
\sin^2 x = \sin^2 x
\]

This is a true statement, so the original equation is **verified**.

Would you like more details or have any other questions?

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Here are five related questions to explore:

1. How do you derive the identity \(\cot x = \frac{\cos x}{\sin x}\)?
2. What is the relationship between \(\sin^2 x\) and \(\cos^2 x\)?
3. How do you simplify expressions involving trigonometric identities?
4. What are the key Pythagorean identities in trigonometry?
5. Can you explain how \(\cot^2 x + 1 = \csc^2 x\) is derived?

**Tip**: Mastering fundamental trigonometric identities will make complex expressions easier to manage!

cot^2x · sin^2x + sin^2x - cos^2x = sin^2x

This page provides a step-by-step solution to the trigonometric identity cot^2x · sin^2x + sin^2x - cos^2x = sin^2x. The problem explores key trigonometric concepts like cotangent and sine, and uses important identities such as cot(x) = cos(x) / sin(x) and the Pythagorean identity. Suitable for high school students in Grades 10-12.

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