Prove the formulae(4) to (7).
`d/dx(cos^-1x)=(-1)/(sqrt1-x^2)`

Prove that tan^(-1)sqrt(x) =(1)/(2) cos^(-1)((1-x)/(1+x)), x in (0, 1) .

Prove that (d)/(dx)[(sin2x+1/(cos^2x)]

prove that (d)/(dx)[(1-tanx)/(1+tanx)]^(1/2)=(-1)/((sqrt(cos2x))cosx)

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

Write the value of d/dx(sin^-1x+cos^-1x) for x in(-1,1) .

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

Prove the following "tan"^(-1)((1-x)/(1+x))-"tan"^(-1)((1-y)/(1+y))="sin"^(-1)((y-x)/(sqrt(1+x^(2))sqrt(1+y^(2)))) .

Find x, for which d/dx cos(cos^-1 x)=1 .

Differentiate (x sin^(-1) x)/(sqrt (1 + x^2))

Prove that cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=(x)/(2), 0 lt x lt (pi)/(2), or x in (0, (pi)/(4)) .