Question

Simplify: `cos^-1x + cos^-1 [x/2 + sqrt(3 - 3x^2)/2];  1/2 ≤ x ≤ 1`

Answer 1

`cos^-1x + cos^-1 [x/2 + sqrt(3 - 3x^2)/2]'  1/2 ≤ x ≤ 1`

Let `cos^-1 x = alpha, x = cos alpha`

Now, = `alpha + cos^-1 [(cos alpha)/2 + (sqrt(3 -3 cos^2 alpha))/2]`

= `alpha + cos^-1 [cos alpha/2 + (sqrt3 .sqrt(1 - cos^2 alpha))/2]`

= `alpha + cos^-1 [cos alpha/2 + sqrt3/2 sin alpha]`

= `alpha + cos^-1 [cos  pi/3 cos alpha + sin  pi/3 sin alpha]`

= `alpha + cos^-1 [cos(pi/3 - alpha)]`

= `pi/3`