# The value of expression (2(sin1+sin2+...+sin89))/(2(cos1+cos2+...+cos44)+1)=csctheta, theta in (0,pi/2) then find costheta and tantheta?

Jul 17, 2018

The value of expression $\frac{2 \left(\sin 1 + \sin 2 + \ldots + \sin 89\right)}{2 \left(\cos 1 + \cos 2 + \ldots + \cos 44\right) + 1} = \csc \theta , \theta \in \left(0 , \frac{\pi}{2}\right)$ then find $\cos \theta \mathmr{and} \tan \theta$

Numerator of LHS

$= 2 \left(\sin 1 + \sin 2 + \ldots + \sin 89\right)$

$= 2 \left(2 \sin 45 \cos 44 + 2 \sin 45 \cos 43 + 2 \sin 45 \cos 42 + \ldots + 2 \sin 45 \cos 1 + \sin 45\right)$

$= \sqrt{2} \left(2 \left(\cos 44 + \cos 43 + \cos 42 + \ldots + \cos 1\right) + 1\right)$

Now

$\frac{\sqrt{2} \left(2 \left(\cos 44 + \cos 43 + \cos 42 + \ldots + \cos 1\right) + 1\right)}{2 \left(\cos 1 + \cos 2 + \ldots + \cos 44\right) + 1} = \csc \theta$

$\implies \csc \theta = \sqrt{2} = \csc 45$

$\implies \theta = 45$

So $\cos \theta = \cos 45 = \frac{1}{\sqrt{2}}$

And

$\tan \theta = \tan 45 = 1$