Question

The number of solutions of the equation \(1 + \sin x \cdot \sin^2 \frac{x}{2} = 0\) in \([-\pi, \pi]\) is

• A. zero
• B. 1
• C. 2
• D. 3

Hint:

Check the range of both sides of the equation.

Answer 1

The Correct Option is A

Step 1: Formula / Definition}
\[ \sin^2 \frac{x}{2} = \frac{1 - \cos x}{2} \]
Step 2: Calculation / Simplification}
\(1 + \sin x \cdot \frac{1 - \cos x}{2} = 0\)
\(2 + \sin x - \sin x \cos x = 0\)
\(2 + \sin x - \frac{1}{2}\sin 2x = 0\)
\(4 + 2\sin x - \sin 2x = 0\)
\(\sin 2x = 2\sin x + 4\)
LHS \(\leq 1\), RHS \(\geq 2\) (since \(2\sin x \geq -2\))
No solution.
Step 3: Final Answer
\[ 0 \]