if eight theta is equal to pi show that cos7 theta+ cot theta is equal to zero

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)