Question

Simplified expression of \[ 1 - \frac{\sin^2 y}{1 + \cos y} + \frac{1 + \cos y}{\sin y} - \frac{\sin y}{1 - \cos y} \]

• A. \( \sin y \)
• B. \( \cos y \)
• C. 1
• D. 0

Hint:

When simplifying trigonometric expressions, use standard identities like \( \sin^2 y + \cos^2 y = 1 \) to eliminate terms and simplify the expression.

Answer 1

The Correct Option is B

Step 1: Understand the expression.
The expression is:
\[ 1 - \frac{\sin^2 y}{1 + \cos y} + \frac{1 + \cos y}{\sin y} - \frac{\sin y}{1 - \cos y} \]
We need to simplify this trigonometric expression.

Step 2: Simplify the first term.
The first term is \( 1 - \frac{\sin^2 y}{1 + \cos y} \). We can rewrite \( 1 - \frac{\sin^2 y}{1 + \cos y} \) as:
\[ 1 - \frac{\sin^2 y}{1 + \cos y} = \frac{(1 + \cos y) - \sin^2 y}{1 + \cos y} \]
Now, using the Pythagorean identity \( \sin^2 y + \cos^2 y = 1 \), we replace \( \sin^2 y \) with \( 1 - \cos^2 y \). This gives:
\[ \frac{1 + \cos y - (1 - \cos^2 y)}{1 + \cos y} = \frac{\cos^2 y + \cos y}{1 + \cos y} \]
This simplifies further to: \[ \frac{\cos y (\cos y + 1)}{1 + \cos y} \]
Now, cancel out \( (1 + \cos y) \) in the numerator and denominator, leaving: \[ \cos y \]

Step 3: Simplify the second term.
The second term is \( \frac{1 + \cos y}{\sin y} \). This term is already simplified, so it remains as:
\[ \frac{1 + \cos y}{\sin y} \]

Step 4: Simplify the third term.
The third term is \( - \frac{\sin y}{1 - \cos y} \), which is also already in a simplified form.

Step 5: Combine the terms.
Now, putting the simplified terms together:
\[ \cos y + \frac{1 + \cos y}{\sin y} - \frac{\sin y}{1 - \cos y} \]
We recognize that the expression is quite complex, but further simplification will reveal that the expression simplifies to \( \cos y \) due to the cancellation of terms and trigonometric identities.

Step 6: Conclusion.
Thus, the simplified expression is: \[ \boxed{\cos y} \]