Question

The value of the expression $\frac{2\left(\sin 1^{\circ }+\sin 2^{\circ }+\sin 3^{\circ }+..+\sin 89^{\circ }\right)}{2\left(\cos 1^{\circ }+\cos 2^{\circ }+...+\cos 44^{\circ }\right)+1}$ 1s equal to

• A. $\sqrt{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{2}$
• D. 2

Answer 1

Answer: (A)

Given expression
$=\frac{2\left\{\left(\sin 1^{\circ }+\sin 89^{\circ }\right)+\left(\sin 2^{\circ }+\sin 88^{\circ }\right)+......+\left(\sin 44^{\circ }+\sin 46^{\circ }\right)+\sin 45^{\circ }\right\}}{2\left(\cos 1^{\circ }+\cot 2^{\circ }+...+\cos 44^{\circ }\right)+1}$
$=\frac{2\left\{2\sin 45^{\circ }\cos 44^{\circ }+2\sin 45^{\circ }\cos 43^{\circ }+...+2\sin 45^{\circ }\cos 1^{\circ }+\sin 45^{\circ }\right\}}{2\left(\cot 1^{\circ }+\cos 2^{\circ }+....+\cos 44^{\circ }\right)+1}$
$=\frac{2\sin 45^{\circ }\left\{2\left(\cos 44^{\circ }+\cos 43^{\circ }+...+\cos 1^{\circ }\right)+1\right\}}{2\left(\cos 1^{\circ }+\cos 2^{\circ }+...+\cos 44^{\circ }\right)+1}$
$=2\sin 45^{\circ }=2\times \frac{1}{\sqrt{2}}=\sqrt{2}$