`L e tf(n)=1+1/2+1/3++1/ndot` Then show that `f(n)=int_0^(pi/2)cot(theta/2)(1-cos^ntheta)d theta`

(1+tan^(2)theta)/(1+cot^(2)theta)=

Show that (cosec theta -cot theta )^(2) =(1-cos theta )/( 1+cos theta )

(1-cos^(2)theta)(1+cot^(2)theta)= ………. .

Let f beintegrable over [0,a] for any real value of a . If I_(1)=int_(0)^(pi//2)cos theta f(sin theta +cos^(2) theta) d theta and I_(2)=int_(0)^(pi//2) sin 2 theta f(sin theta+cos^(2) theta) d theta , then

Let n ge 2 be a natural number and 0 lt theta lt (pi)/(2) , Then, int ((sin^(n)theta - sin theta)^(1/n) cos theta)/(sin^(n+1) theta)d theta is equal to (where C is a constant of integration)

Iflambda=int_0^(pi/2)costhetaf(sintheta+cos^2theta)dtheta a n dI_2=int_0^(pi//2)sin2thetaf(sintheta+cos^2theta)dtheta,t h e n I_1=-2I_2 (b) I_1=I_2 2I_1=I_2 (d) I_1=-I_2

Simplify: cos^(2)theta+(1)/(1+cot^(2)theta):

If U_n=int_0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zero, then show that U_(n+2)+U_n=2U_(n+1)dot Hence, deduce that int_0^(pi/2)(sin^2ntheta)/(sin^2theta)=1/2npidot