Maximum value of `sum_(n=1)^3(cos^(2n)theta+(-1)^(n+1)sin^(2n)theta),AAtheta in R in R` is -

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 ]

sum_(i=1)^(n)cos theta_(i)=n then sum_(i=1)^(n)sin theta_(i)

Maximum value of n for which sum_(r=1)^(n+15)1>sum_(r=1)^(n)(r+(1)/(2)) is

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

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