Question

The smallest integer \( n \) such that \[ \frac{1}{\sin 45^\circ \sin 46^\circ} + \frac{1}{\sin 47^\circ \sin 48^\circ} + \cdots + \frac{1}{\sin 133^\circ \sin 134^\circ} = \frac{1}{\sin(n^\circ)} \]

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

Trigonometric Series Symmetry}
Use angle identities: \( \sin(180^\circ - x) = \sin x \)
Group terms like \( \frac{1}{\sin x \sin(180^\circ - x)} \)
Look for patterns in consecutive angle pairings

Answer 1

The Correct Option is A

Use the identity: \[ \frac{1}{\sin x \sin(180^\circ - x)} = \frac{1}{\sin x \sin x} = \frac{1}{\sin^2 x} \] Group terms symmetrically around \( 90^\circ \) — terms like: \[ \frac{1}{\sin 45^\circ \sin 134^\circ}, \frac{1}{\sin 46^\circ \sin 133^\circ}, \ldots \] These reduce using symmetry and ultimately sum up to: \[ \frac{1}{\sin 1^\circ} \]