Question

The value of the expression $\frac{2(sin{1}^{∘}+sin{2}^{∘}+sin{3}^{∘}+........+sin89^{∘})}{2(cos{1}^{∘}+cos{2}^{∘}+cos{3}^{∘}+........+cos44^{∘})+1}$ is equal to

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

Answer 1

Answer: (A)

$\frac{2\left\{\left(\sin 1^{∘}+\sin 89^{∘}\right)+\left(\sin 2^{∘}+\sin 88^{∘}\right)+……+\left(\sin 44^{∘}+\sin 46^{∘}\right)+\sin 45^{∘}\right\}}{2\left(\cos 1^{∘}+\cos 2^{∘}+\cos 3^{∘}+……+\cos 44^{∘}\right)+1}$
$=\frac{2\left\{2\sin 45^{∘}\left(\cos 44^{∘}+\cos 43^{∘}+……+\cos 1^{∘}\right)+\sin 45^{∘}\right\}}{2\left(\cos 1^{∘}+\cos 2^{∘}+\cos 3^{∘}+……+\cos 44^{∘}\right)+1}$
$=\frac{2\sin 45^{∘}\left\{2\left(\cos 1^{∘}+\cos 2^{∘}+……+\cos 44^{∘}\right)+1\right\}}{2\left(\cos 1^{∘}+\cos 2^{∘}+……+\cos 44^{∘}\right)+1}$
$=2â‹\dots \frac{1}{\sqrt{2}}$
$=\sqrt{2}$