Question

The value of expression $\frac{2(\sin 1^0+\sin 2^0+\sin 3^0+........+\sin {89}^0)}{2(\cos 1^0+\cos 2^0+.......+\cos {44}^0)+1}$ equals

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

Answer 1

The correct option is B $\sqrt{2}$

$\frac{2\left(\sin 1°+\sin 2°+sin3°+..........\sin 89°\right)}{2\left(\cos 1°+\cos 2°+.........+\cos 44°\right)+1}\sin \theta =\cos (90-\theta )\frac{2\left(cos1°+cos2°+............cos89°\right)}{2\left(\cos 1°+\cos 2°+...........\cos 44°\right)+1}As\cos A+\cos B=2cos\frac{A+B}{2}\cos \frac{A-B}{2}\cos 1°+\cos 89°=2\cos \frac{90}{2}\cos \frac{88}{2}=2\cos 45°\cos 44°\Rightarrow \cos 2°+\cos 88°=2\cos 45°\cos 43°Wehave,\frac{2\left(2\cos 45°\cos 44°+2\cos 45°\cos 43°+............+2\cos 45°\cos 1°+\cos 45°\right)}{2\left(\cos 1°+\cos 2°+..............+\cos 44°\right)+1}2\cos 45°=\frac{2}{\sqrt{2}}=\sqrt{2}$.