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

Prove that, (sin (x/2)-cos (x/2))^(2)=1-sinx

Prove that sin^(-1) cos (sin^(-1) x) + cos^(-1) x) = (pi)/(2), |x| le 1

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

Prove that sin^(-1). ((x + sqrt(1 - x^(2))/(sqrt2)) = sin^(-1) x + (pi)/(4) , where - (1)/(sqrt2) lt x lt(1)/(sqrt2)

Prove that sqrt ((1- cos 2x)/( 1 + cos 2x )) = tan x, x in I or III quad.

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

Prove that sqrt(sin^4x+4cos^2x)-sqrt(cos^4x+4sin^2x)=cos2xdot

Prove that (1+cos x)/(sin x) = (cos( x/2))/(sin (x/2))

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

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