# if eight theta is equal to pi show that cos  7 theta+ cot theta is equal to zero

Arun
25750 Points
6 years ago
8*theta=pi, so theta=pi/8.
Now, from half angle formula cos(theta/2) =± sqrt((1+ cos theta)/2).
Here pi/8 = ½ (pi/4) so we can apply half angle formula and
get cos(pi/8) = +sqrt((1+cos pi/4)/2)
= sqrt((1+(1/sqrt 2))/2
)=sqrt((2+sqrt2)/2) –(1)
Similarly for 7pi/8, applying half angle formula.
Here 7pi/8 = ½ (7pi/4) and
cos(7pi/4)=cos(2pi-pi/4)
=cos(pi/4
)=1/sqrt(2)
So cos(7pi/8) = -sqrt((1+cos 7pi/4)/2)[negative as 7pi/8 lies in second quadrant where cos is negative] =
-sqrt((1+(1/sqrt 2))/2)=
-sqrt((2+sqrt2)/2)—(2)

therefore , from (1) and (2) we get cos 7pi/4 + cos pi/4 =0
or cos 7theta + cos theta=0 (proved)
Soumendu Majumdar
159 Points
6 years ago
theta=pi/8
so cos 7theta = cos(pi-pi/8) = -cos(pi/8) =-root{(1 + cos(pi/4))/2} =-root{(2root2 + 1)/2root2}
now cos theta = cos pi/8 =root{(2root2 + 1)/2root2}
therefore cos7theta + cos theta =0
P.S--The question is wrong...it would be cos theta instead of cot theta!
Hope it helps!