If `xsum_(n=0)^(oo)cos^(2n)theta,y=sum_(n=0)^(oo)sin^(2n)thetaandz=sum_(n=0)^(oo)cos^(2n)thetasin^(2n)theta,ltthetalt(pi)/(2),` then show that xyz = x + y + z.

If x = sum_(n = 0)^(infty) cos ^(2n) theta, " "y = sum_(n = 0)^(infty) sin^(2n) theta and z = sum_(n = 0)^(infty) cos^(2n) theta sin^(2n)theta, 0 lt theta lt (pi)/(2) , then show that xyz = x + y + z [ Hint : use the formula 1 + x + x^(2) + x^(3) +... =(1)/(1 - x) , where |x| lt 1 ]

If 0 lt phi lt (pi)/(2) , x= sum_(n=0)^(oo) cos^(2n) phi ,y sum _(n=0)^(oo) sin ^(2n) phi and z = sum _(n=0) ^(oo) cos ^(2n) phi sin ^(2n) phi , then

Find sum_(n=1)^(oo)1/(n^2+5n+6)

If theta in (pi//4, pi//2) and sum_(n=1)^(oo)(1)/(tan^(n)theta)=sin theta + cos theta , then the value of tan theta is

sum_(n=1)^ootan(theta/(2^n))/(2^(n-1)cos(theta/(2^(n-1))))

sin^2 n theta- sin^2 (n-1)theta= sin^2 theta where n is constant and n != 0,1

Find the sum_(k=1)^(oo) sum_(n=1)^(oo)k/(2^(n+k)) .