Find tancos123 sinsec1xSolutiontancos123 sinsec1x Let y cos1

Find tan(cos^-1(2/3)) sin(sec^-1(x))

tan(cos^-1(2/3))* sin(sec^-1(x))

Let y =cos^-1(2/3)

cos y = 2/3 ; tany = sqrt(5)/2

tan(cos^-1(2/3)) = tan(tan^-1(sqrt5/2)) = sqrt(5)/2

sin(sec^-1(x))

Let z = sec^-1(x) ; secz = x

cosz = 1/x ; sinz = sqrt(1-x^2)/x

z = sin^-1(sqrt(1-x^2)/x)

So , sin(sec^-1(x)) = sinsin^-1(sqrt(1-x^2)/x)

= sqrt(1-x^2)/x

Finally , tan(cos^-1(2/3))* sin(sec^-1(x))

= sqrt(5)/2*sqrt(1-x^2)/x

= sqrt(5 -5x^2)/2x