Question

The value of \( \frac{1}{2\sin 10^\circ} - 2\sin 70^\circ \) is

• A. 0
• B. 2
• C. \( -\frac{3}{2} \)
• D. 1

Hint:

Use identities like \( \sin(90^\circ-\theta)=\cos\theta \) and \( 2\sin A \cos A = \sin 2A \) to simplify trigonometric expressions.

Answer 1

The Correct Option is D

Step 1: Use trigonometric identity.
Recall that:
\[ \sin(90^\circ - \theta) = \cos \theta. \]
So,
\[ \sin 70^\circ = \cos 20^\circ. \]

Step 2: Rewrite the expression.
\[ \frac{1}{2\sin 10^\circ} - 2\cos 20^\circ. \]

Step 3: Use identity for cosine.
\[ \cos 20^\circ = \frac{\sin 40^\circ}{2\sin 20^\circ}. \]
Alternatively, use identity:
\[ 2\sin A \cos A = \sin 2A. \]

Step 4: Apply identity to simplify.
We use:
\[ \sin 20^\circ = 2\sin 10^\circ \cos 10^\circ. \]
So,
\[ \frac{1}{2\sin 10^\circ} = \frac{\cos 10^\circ}{\sin 20^\circ}. \]

Step 5: Express everything in terms of \( \sin 20^\circ \).
Also:
\[ 2\cos 20^\circ = \frac{2\cos 20^\circ \sin 20^\circ}{\sin 20^\circ}. \]
\[ = \frac{\sin 40^\circ}{\sin 20^\circ}. \]

Step 6: Combine the terms.
\[ \frac{\cos 10^\circ}{\sin 20^\circ} - \frac{\sin 40^\circ}{\sin 20^\circ}. \]
Using identity:
\[ \sin 40^\circ = 2\sin 20^\circ \cos 20^\circ. \]
After simplification, expression reduces to:
\[ 1. \]

Step 7: Final conclusion.
Thus, the required value is 1.
Final Answer:
\[ \boxed{1}. \]