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

1 Answer
Jul 17, 2018

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

Numerator of LHS

=2(sin1+sin2+...+sin89)

=2(2sin45cos44+2sin45cos43+2sin45cos42+...+2sin45cos1+sin45)

=sqrt2(2(cos44+cos43+cos42+...+cos1)+1)

Now

(sqrt2(2(cos44+cos43+cos42+...+cos1)+1))/(2(cos1+cos2+...+cos44)+1)=csctheta

=>csctheta=sqrt2=csc45

=>theta=45

So costheta=cos45=1/sqrt2

And

tantheta=tan45=1