" (D) "sin{2tan^(-1)(sqrt((1)/(1+x)))}" (Hint "x=cos theta)

y=tan^(-1)((x)/(1+sqrt(1-x^(2))))+sin(2tan^(-1)theta*sqrt((1-x)/(1+x))) then prove that ,4(1-x^(2))^(3)((d^(2)y)/(dx^(2)))^(2)+4x=x^(2)+4

tan ( 2 tan ^(-1)sqrt((1+cos theta)/(1-cos theta)))+ tan theta =0

prove that , tan(2tan ^(-1) sqrt((1+cos theta )/(1- cos theta )))+tan theta =0

Prove that: "sin"[cot^(-1){"cos"(tan^(-1)x)}]=sqrt((x^2+1)/(x^2+2)) cos [tan^(-1) (cot^(-1)x)}]=sqrt((x^2+1)/(x^2+2))

int(x)/(sqrt(1-x))dx=int(sin^(2)theta)/(sqrt(1-sin^(2)theta))*2sin theta cos theta d theta

tan(2tan^(-1)sqrt((1+cos theta)/(1-cos theta)))+tan theta=0

f(x)=cos(2tan^-1(sin(cot^-1sqrt((1-x)/x)))) then

If x=5+2 sqrt(6) and tan theta=(1)/(2) ( sqrt(x)+(1)/(sqrt(x))) then sec^(2) theta+ sin^(2) theta=

(1) / (2) cos ^ (- 1) x = sin ^ (- 1) sqrt ((1-x) / (2)) = cos ^ (- 1) sqrt ((1 + x) / (2 )) = (tan ^ (- 1) (sqrt (1-x ^ (2)))) / (1 + x)