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

Options

A. `(pi)/(3)`
B. `2 cos^(-1) 2/3 -(pi)/(3)`
C. `(2pi)/(3)`
D. `2cso^(-1)2/3`

Explanation

We have
`cos^(-1){xy+sqrt(1-x^(2))sqrt(1-y^(2))}`
`={:(cos^(-1)x-cosf^(-1)y, if -1 le x,y le 1 and x le y),(cos^(-1)y-cos^(-1)x if -1 le y le 0, 0 le x le 1 and y le x):}`
` rarr f(2/3)= cos^(-1)1/2=(pi)/(3)`