Question

The value of the expression $\frac{2\left(\sin1^{\circ} + \sin2^{\circ} +\sin3^{\circ} +..+\sin89^{\circ}\right)}{2\left(\cos1^{\circ} +\cos 2^{\circ} + ...+\cos44^{\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} +\sin89^{\circ}\right) + \left(\sin2^{\circ} + \sin88^{\circ}\right) + ......+\left(\sin44^{\circ} + \sin46^{\circ}\right) + \sin45^{\circ}\right\}}{2\left(\cos1^{\circ} +\cot2^{\circ} + ...+ \cos 44^{\circ} \right) + 1} $
$ = \frac{2\left\{2 \sin45^{\circ} \cos44^{\circ} +2\sin45^{\circ} \cos43^{\circ} +... +2 \sin45^{\circ} \cos1^{\circ}+\sin45^{\circ}\right\}}{2\left(\cot 1^{\circ} + \cos2^{\circ} + ....+\cos44^{\circ} \right) + 1 } $
$ = \frac{2\sin45^{\circ} \left\{2\left(\cos 44^{\circ } +\cos43^{\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}$