Question

The value of the expression $\frac{2 \left(\right. s i n 1 ^\circ + s i n 2 ^\circ + s i n 3 ^\circ + . . . . . . . . + s i n 89 ^\circ \left.\right)}{2 \left(\right. c o s 1 ^\circ + c o s 2 ^\circ + c o s 3 ^\circ + . . . . . . . . + c o s 44 ^\circ \left.\right) + 1}$ is equal to

• A. $\sqrt{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{2}$
• D. 0

Answer 1

Answer: (A)

$\frac{2\left\{\left(\sin 1^{\circ}+\sin 89^{\circ}\right)+\left(\sin 2^{\circ}+\sin 88^{\circ}\right)+\ldots \ldots+\left(\sin 44^{\circ}+\sin 46^{\circ}\right)+\sin 45^{\circ}\right\}}{2\left(\cos 1^{\circ}+\cos 2^{\circ}+\cos 3^{\circ}+\ldots \ldots+\cos 44^{\circ}\right)+1}$
$=\frac{2\left\{2 \sin 45^{\circ}\left(\cos 44^{\circ}+\cos 43^{\circ}+\ldots \ldots+\cos 1^{\circ}\right)+\sin 45^{\circ}\right\}}{2\left(\cos 1^{\circ}+\cos 2^{\circ}+\cos 3^{\circ}+\ldots \ldots+\cos 44^{\circ}\right)+1}$
$=\frac{2 \sin 45^{\circ}\left\{2\left(\cos 1^{\circ}+\cos 2^{\circ}+\ldots \ldots+\cos 44^{\circ}\right)+1\right\}}{2\left(\cos 1^{\circ}+\cos 2^{\circ}+\ldots \ldots+\cos 44^{\circ}\right)+1}$
$=2 \cdot \frac{1}{\sqrt{2}}$
$=\sqrt{2}$