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To prove the trigonometric identity:

\[
\frac{\sin^2 x}{1 + \cos x} = 1 - \cos x
\]

Let's work through the steps to prove this. The strategy involves transforming the left-hand side (LHS) into the right-hand side (RHS) by using known trigonometric identities.

### Step-by-step Proof:

#### Step 1: Express \(\sin^2 x\) using the Pythagorean identity.
We know that:
\[
\sin^2 x = 1 - \cos^2 x
\]
So, substitute this into the LHS:

\[
\frac{1 - \cos^2 x}{1 + \cos x}
\]

#### Step 2: Factor the numerator.
Notice that \(1 - \cos^2 x\) is a difference of squares, which factors as:
\[
1 - \cos^2 x = (1 - \cos x)(1 + \cos x)
\]
Now substitute this into the equation:

\[
\frac{(1 - \cos x)(1 + \cos x)}{1 + \cos x}
\]

#### Step 3: Simplify the fraction.
Since \(1 + \cos x\) appears in both the numerator and the denominator, we can cancel it out, assuming \(1 + \cos x \neq 0\):

\[
1 - \cos x
\]

This is exactly the right-hand side (RHS) of the equation, so the identity is proven.

\[
\boxed{\frac{\sin^2 x}{1 + \cos x} = 1 - \cos x}
\]

---

Would you like further details on this proof or any clarifications?

### Related Questions:
1. Can you explain the Pythagorean identity in trigonometry?
2. How does the difference of squares factorization work?
3. What are the conditions under which you can cancel terms in a fraction?
4. How would you prove this identity using a different method?
5. What happens if \( \cos x = -1 \)?

#### Tip:
Always check for any restrictions on the variables when simplifying trigonometric expressions, such as dividing by zero.

Prove the identity: sin^2(x) / (1 + cos(x)) = 1 - cos(x)

Learn how to prove the trigonometric identity sin^2(x) / (1 + cos(x)) = 1 - cos(x) using the Pythagorean identity and difference of squares. Step-by-step breakdown for Grades 10-12 students.

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