Prove that:`tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)), x in [0,1]`

Prove the following: tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)),\ x\ \ (0,1)

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

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

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)[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2cos^(-1)x ,\ -1/(sqrt(2))\ lt=x\ lt=1

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

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

Prove that 1/2 cos^(-1) ((1-x)/(1+x))=tan^(-1) sqrtx .

Prove that: tan^(-1){(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))}=pi/4-1/2. cos^(-1)x , 0

Prove that: tan^(-1) {(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))} = pi/4-1/2\ cos^(-1)x