Which of the following expression is equivalent to `(sin^(2)x)/(1+cosx)`?

Options

A. `1-sinx`
B. `1-cosx`
C. `sinx+cosx`
D. `sinx-cosx`

Answer

To solve the expression \(\frac{\sin^2 x}{1 + \cos x}\), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 x + \cos^2 x = 1 \]
From this, we can express \(\sin^2 x\) in terms of \(\cos^2 x\):
\[ \sin^2 x = 1 - \cos^2 x \]

### Step 2: Substitute \(\sin^2 x\) in the Expression
Now, we substitute \(\sin^2 x\) into the original expression:
\[ \frac{\sin^2 x}{1 + \cos x} = \frac{1 - \cos^2 x}{1 + \cos x} \]

### Step 3: Factor the Numerator
The numerator \(1 - \cos^2 x\) can be factored using the difference of squares:
\[ 1 - \cos^2 x = (1 - \cos x)(1 + \cos x) \]
Thus, we can rewrite our expression as:
\[ \frac{(1 - \cos x)(1 + \cos x)}{1 + \cos x} \]

### Step 4: Cancel Common Factors
Now, we can cancel the common factor \((1 + \cos x)\) in the numerator and denominator:
\[ \frac{(1 - \cos x)(1 + \cos x)}{1 + \cos x} = 1 - \cos x \]

### Final Result
Therefore, the expression \(\frac{\sin^2 x}{1 + \cos x}\) simplifies to:
\[ 1 - \cos x \]