show that cos2thetacos(theta/2)-cos3thetacos((9theta)/2)=sin5thetasin((5theta)/2)

Prove that costhetacos""(theta)/(2)-cos3 thetacos""(9theta)/(2)=sin7thetasin8theta .

Prove that ( cos theta + cos 2 theta + cos 3 theta + cos 4 theta)/(sin theta + sin 2 theta + sin 3 theta + sin 4 theta)= cot ""(5theta)/(2).

(sin theta)/(cos(3theta))+(sin(3theta))/(cos(9theta))+(sin(9theta))/(cos(27theta))+(sin(27theta))/(cos(81theta))=

If (cos theta_(1))/(cos theta_(2))+(sin theta_(1))/(sin theta_(2))=(cos theta_(0))/(cos theta_(2))+(sin theta_(0))/(sin theta_(2))=1 , where theta_(1) and theta_(0) do not differ by can even multiple of pi , prove that (cos theta_(1)*cos theta_(0))/(cos^( 2)theta_(2))+(sin theta_(1)*sin theta_(0))/(sin^(2) theta_(2))=-1

Prove that : sintheta/cos(3theta)+sin(3theta)/cos(9theta)+sin(9theta)/cos(27theta)=1/2(tan(27theta)-tan(theta))

1 + (cos 2 theta + cos 6 theta)/(cos 4 theta) = (sin 3 theta)/(sin theta).

int (d theta)/((sin theta - 2 cos theta)(2 sin theta + cos theta))

If theta-phi=pi/2, prove that, [(cos^2 theta,cos theta sin theta),(cos theta sin theta,sin^2 theta)] [(cos^2 phi,cos phi sin phi),(cos phi sin phi,sin^2 phi)]=0

If theta-phi=pi/2, prove that, [(cos^2 theta,cos theta sin theta),(cos theta sin theta,sin^2 theta)] [(cos^2 phi,cos phi sin phi),(cos phi sin phi,sin^2 phi)]=0