Prove that `cos^(-1){(1+x)/2}=(cos^(-1)x)/2`

If |x|<=1 then prove that cos^(-1)(-x)=pi-cos^(-1)x

Prove that cos^(-1)(3x-4x^3)=3cos^(-1)x,x in[1/2,1]

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

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

Prove that: 3cos^(-1)x=cos^(-1)(4x^(3)-3x),x in[(1)/(2),1]

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

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

Prove that i) cos^(-1)(1-2x^(2))=2sin^(-1)x ii) cos^(-1)(2x^(2)-1)=2cos^(-1)x . iii) sec^(-1)(1/(2x^(2)-1)=2cos^(-1)x iv) cot^(-1)(sqrt(1-x^(2))-x)=pi/2-1/2cot^(-1)x .

Prove that cos^(-1)x+cos^(-1) [x/2 +(sqrt(3-3x^(2)))/2 ] = pi/3 , 1/2 le x le 1

prove that sin^(-1)(x)+cos^(-1)(x)=pi/2 , -1<=x<=1