Find the range of `f(x)=sin^(-1)x+tan^(-1)x+cos^(-1)xdot`

Answer

`[(pi)/(4), (7pi)/(4)]`

Explanation

We have `f(x) = cos^(-1) x + cot^(-1) x`
Clearly, domain of the function is `[-1 ,1]`
Now, both `cos^(-1) x and cot^(-1) x` are decreasing function
`:.` f(x) is also decreasing function
`:.` Range of the function is `[f (1), f(-1)]`
Now, `f(-1) = cos^(-1) + cot^(-1) (-1) = pi + 3 pi//4 = 7 pi//4`
`f(1) = cos^(-1) (1) + cot^(-1) (1) = 0 + pi//4 = pi//4`
Hence, range is `[pi//4, 7pi//4]`