[" Prove that if "(1)/(2)leqslant x<=1" then "],[cos^(-1)x+cos^(-1)[(x)/(2)+(sqrt(3-3x^(2)))/(2)]=pi/3]

If x be real then prove that (x-1)(x-2)+1 is always positive.

If f(x)=(x^(2)-1)/(x^(2)+1) prove that f((1)/(x))=-f(x)

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

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

For x>0 Prove that x-(x^(2))/(2)

If e^(cot^(-1)x) Prove that (1+x^(2))y_(2) +(2x + 1)y_(1) = 0 .

i) Prove that: (1-sin2A)/(1+sin2A) = tan^(2)(pi/4-A) ii) If costheta=1/2(x+1/x) , then prove that: cos2theta=1/2(x^(2)+1/x^(2)) and cos3theta=1/2(x^(3)+1/x^(3))

i) Prove that: (1-sin2A)/(1+sin2A) = tan^(2)(pi/4-A) ii) If costheta=1/2(x+1/x) , then prove that: cos2theta=1/2(x^(2)+1/x^(2)) and cos3theta=1/2(x^(3)+1/x^(3))

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

If 2x=secA and (2)/(x)=tanA, " prove that " (x^(2)-(1)/(x^(2)))=(1)/(4).