Prove that `tan^(-1){x/(a+sqrt(a^2-x^2))}=1/2sin^(-1)x/a ,-a lt x lt a`

Prove that tan^(-1) {(x)/(a + sqrt(a^(2) - x^(2)))} = (1)/(2) sin^(-1).(x)/(a), -a lt x lt a

Prove that cos^(-1) {sqrt((1 + x)/(2))} = (cos^(-1) x)/(2) , -1 lt x lt 1

Differentiate the following function with respect to x : tan^(-1){x/(sqrt(a^2-x^2))} , -a lt xlt a

tan^(-1)""(x)/(sqrt(a^(2)-x^(2))),|x|lt a

Prove that: sin^(-1){(sqrt(1+x)+sqrt(1-x))/2}=pi/2-(sin^(-1)x)/2,""0 < x < 1

Prove that sin^(-1). ((x + sqrt(1 - x^(2))/(sqrt2)) = sin^(-1) x + (pi)/(4) , where - (1)/(sqrt2) lt x lt(1)/(sqrt2)

Prove that tan^(-1)(sqrt((1-cosx)/(1+cosx))=x/2, x lt pi .

Show that(i) sin^(-1)(2xsqrt(1-x^2))=2sin^(-1)x ,-1/(sqrt(2))lt=xlt=1/(sqrt(2)) (ii) sin^(-1)(2xsqrt(1-x^2))=2cos^(-1)x ,1/(sqrt(2))lt=xlt=1

Prove that sin^(-1)((2x)/(1+x^2))=tan^(-1)((2x)/(1-x^2))

Differentiate the following function with respect to x : sin^(-1){(x+sqrt(1-x^2))/(sqrt(2))} , -1 lt x lt1