[quad sin^(-1)(sum_(i=1)^(oo)x^(i+1)-x sum_(i=1)^(oo)((x)/(2))^(i))=(pi)/(2)-cos^(-1)(sum_(i=1)^(oo)(-(x)/(2))^(i)-sum_(i=1)^(oo)(-x)^(i))" Iying in the interval "(-(1)/(2),(1)/(2))" is "],[" (Here,the inverse trigonometric functions "sin^(-1)x" and "cos^(-1)x" assume value in "[-(pi)/(2),(pi)/(2)]" and "[0,pi]" ,"],[" respectively."]

The number of real solution of the equation sin^(-1) (sum_(i=1)^(oo) x^(i +1) -x sum_(i=1)^(oo) ((x)/(2))^(i)) = (pi)/(2) - cos^(-1) (sum_(i=1)^(oo) (-(x)/(2))^(i) - sum_(i=1)^(oo) (-x)^(i)) lying in the interval (-(1)/(2), (1)/(2)) is ______. (Here, the inverse trigonometric function sin^(-1) x and cos^(-1) x assume values in [-(pi)/(2), (pi)/(2)] and [0, pi] respectively)

The number of real solution of the equation sin^(-1) (sum_(i=1)^(oo) x^(i +1) -x sum_(i=1)^(oo) ((x)/(2))^(i)) = (pi)/(2) - cos^(-1) (sum_(i=1)^(oo) (-(x)/(2))^(i) - sum_(i=1)^(oo) (-x)^(i)) lying in the interval (-(1)/(2), (1)/(2)) is ______. (Here, the inverse trigonometric function sin^(-1) x and cos^(-1) x assume values in [-(pi)/(2), (pi)/(2)] and [0, pi] respectively)

The number of real solutions of the equation sin^(-1)(sum_(i=1)^oox^(i+1)-xsum_(i=1)^oo(x/2)^i)=pi/2-cos^(-1)(sum_(i=1)^oo(-x/2)^i-sum_(i=1)^oo(-x)^i) lying in the interval (-1/2,1/2) is ____. (Here, the inverse trigonometric function =sin^(-1)x and cos^(-1)x assume values in [pi/2,pi/2] and [0, pi] , respectively.)

sum_(i=1)^(oo)tan^(-1)((1)/(2r^(2)))=

sum_(i=1)^(oo)tan^(-1)((1)/(2i^(2)))=t, thentant =(B)(1)/(2)

If sum_(r=1)^(oo)(1)/((2r-1)^(2))=(pi^(2))/(8) , then sum_(r=1)^(oo) (1)/(r^(2)) is equal to

sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(2^(i+j+k)) is equal to

If sum_(i=1)^(n)sin x_(i)=n then sum_(i=1)^(n)cot x_(i)=

If sum_(i=1)^(n)sin x_(i)=n then sum_(i=1)^(n)cot x_(i)=