`2cos^-1 x=cos^-1 (2x^2-1)`

Prove that 2 cos^(-1)x =cos^(-1)(2x^(2)-1) .

If f(x)=cos^(-1)x+cos^(-1){(x)/(2)+(1)/(2)sqrt(3-3x^(2))} then

Statement -1: if -1lexle1 then sin^(-1)(-x)=-sin^(-1)x and cos^(-1)(-x)=pi-cos^(-1)x Statement-2: If -1lexlex then cos^(-1)x=2sin^(-1)sqrt((1-x)/(2))= 2cos^(-1)sqrt((1+x)/(2))

Statement -1: if -1lexle1 then sin^(-1)(-x)=-sin^(-1)x and cos^(-1)(-x)=pi-cos^(-1)x Statement-2: If -1lexlex then cos^(-1)x=2sin^(-1)sqrt((1-x)/(2))= 2cos^(-1)sqrt((1+x)/(2))

If 0 < cos^(-1)x < 1 and 1+sin(cos^(-1)x)+sin^2(cos^(-1)x)+sin^3(cos^(-1)x)....+oo=2, then the value of 12 x^2 is____.

If 0

If f(x)=cos^(- 1)x+cos^(- 1){x/2+1/2sqrt(3-3x^2)} then

If f(x)=cos^(- 1)x+cos^(- 1){x/2+1/2sqrt(3-3x^2)} then

Let f(x)=(2cos x-1)(2cos2x-1)(2cos2^(2)x-1)...(2cos2^(n)x-1) (where n>=1). Then prove that f((2 pi k)/(2^(n)+-1))=1AA k in I