(ib)cos^(-1)((x)/(sqrt(x^(2)+a^(2))))

cos^-1((x)/sqrt(1+x^2))

lim _(x to ((1)/(sqrt2))^(+))(cos ^(-1) (2x sqrt(1- x ^(2))))/((x-(1)/(sqrt2)))- lim _(x to ((1)/(sqrt2))^(-))(cos ^(-1) (2x sqrt(1-x ^(2))))/((x- (1)/(sqrt2)))=

cos^(-1)((2sqrt(x))/(1+x))

lim_(x rarr(1)/(sqrt(2)^(+)))(cos^(-1)(2x sqrt(1-x^(2))))/((x-(1)/(sqrt(2))))-lim_(x rarr(1)/(sqrt(2)^(-)))(cos^(-1)(2x sqrt(1-x^(2))))/((x-(1)/(sqrt(2))))

cos^(-1)((x^(2))/(6)+sqrt(1-(x^(2))/(9))sqrt(1-(x^(2))/(4)))=cos^(-1)((x)/(3))-cos^(-1)((x)/(2)) hold for all x belonging to

f(x)=sqrt(sin(cos x))+ln(-2cos^(2)x+3cos x+1)+e^(cos^(-1))((2sin x+1)/(2sqrt(2sin x)))

If 1/(sqrt(2))

If x takes negative permissible values , then sin^(-1) x= a) cos^(-1)sqrt(1-x^2) b) -cos^(-1)sqrt(1-x^2) c) cos^(-1)sqrt(x^2-1) d) pi-cos^(-1)sqrt(1-x^2)

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