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

Prove that 2sin^(-1)x=sin^(-1)[2x sqrt(1-x^(2))]

Prove that sin^(-1)x=cos^(-1) sqrt(1-x^2)

If x lt 0 , then prove that cos^(-1) x = pi + tan^(-1). (sqrt(1 - x^(2)))/(x)

Prove that tan^(-1){(x)/(a+sqrt(a^(2)-x^(2)))}=(1)/(2)(sin^(-1)x)/(a),-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: sin cos^-1 tan sec^-1 x= sqrt(2-x^2)

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

Prove that : tan^(-1)x +tan^(-1). (2x)/(1-x^(2)) = tan^(-1) . (3x-x^(3))/(1-3x^(2)) , |x| lt 1/(sqrt(3))

Prove that tan^(-1) x =sec^(-1) sqrt(1+x^2)

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