Certainly! Let's solve the trigonometric identity [tex]\(\frac{\sin^2 \theta}{1 + \cos \theta} = 1 - \cos \theta\)[/tex] step-by-step.
### Step 1: Understand the Pythagorean Identity
We start by recognizing a well-known Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
From this identity, we can express [tex]\(\sin^2 \theta\)[/tex] in terms of [tex]\(\cos \theta\)[/tex]:
[tex]\[ \sin^2 \theta = 1 - \cos^2 \theta \][/tex]
### Step 2: Substitute [tex]\(\sin^2 \theta\)[/tex] in the Original Equation
Using the Pythagorean identity, we substitute [tex]\(\sin^2 \theta\)[/tex] in the left-hand side (LHS) of the given equation:
[tex]\[ \frac{1 - \cos^2 \theta}{1 + \cos \theta} \][/tex]
### Step 3: Simplify the Expression
To simplify [tex]\(\frac{1 - \cos^2 \theta}{1 + \cos \theta}\)[/tex], observe the numerator:
[tex]\[ 1 - \cos^2 \theta = (1 + \cos \theta)(1 - \cos \theta) \][/tex]
Now, substitute the expanded form into the fraction:
[tex]\[ \frac{(1 + \cos \theta)(1 - \cos \theta)}{1 + \cos \theta} \][/tex]
### Step 4: Cancel Out Common Factors
We see that [tex]\((1 + \cos \theta)\)[/tex] appears in both the numerator and the denominator, so they cancel out:
[tex]\[ \frac{(1 + \cos \theta)(1 - \cos \theta)}{1 + \cos \theta} = 1 - \cos \theta \][/tex]
### Step 5: Verify Equality
After cancellation, we are left with:
[tex]\[ 1 - \cos \theta \][/tex]
which matches exactly with the right-hand side (RHS) of the original equation:
[tex]\[ 1 - \cos \theta \][/tex]
### Conclusion
We have shown that:
[tex]\[ \frac{\sin^2 \theta}{1 + \cos \theta} = 1 - \cos \theta \][/tex]
by substituting [tex]\(\sin^2 \theta = 1 - \cos^2 \theta\)[/tex] and simplifying the resulting expression.
Therefore, the given equation is confirmed as a valid trigonometric identity.
