Find the value of `sin^(-1)(2^x)` (ii) `cos^(-1)sqrt(x^2-x+1)` `tan^(-1)(x^2)/(1+x^2)` (iv) `sec^(-1)(x+1/x)`

Explanation

(i) We have that `2^(@) gt 0`
But for `sin^(-1) (2^(x))` to be defined, we must have `2^(x) le 1`
`:. 0 lt 2^(x) le 1`
`rArr 0 sin^(-1) (2^(x)) le pi//2`
(ii) `cot^(-1) sqrt(x^(2)-x+1)= cos^(-1) sqrt((x-1/2)^(2)+3/4)`
We have `sqrt((x-1/2)^(2)+3/4) ge sqrt(3)/2`
`:. sqrt(3)/2 le sqrt((x-1/2)^(2)+3/4) le 1`
`:. 0 le cos^(-1) sqrt((x-1/2)^(2)+3/4) le pi/6`
(iii) `"tan"^(-1) x^(2)/(1+x^(2))=tan^(-1) (1-1/(1+x^(2)))`
Now, `1 le 1+ x^(2) lt oo`
`:. 0 lt 1/(1+x^(2)) lt 1`
`:. -1 lt -1/(1+x^(2)) lt 0`
`:. 0 lt 1-1/(1+x^(2)) lt 1`
`:. 0 lt tan^(-1) (1-1/(1+x^(2))) lt pi/4`
(iv) Let `x+1/x=y`
`:. x^(2)-yx+1=0`
Since x is real, `D ge 0`
`:. y^(2) -4 ge 0`
`:. y le -2` or `y ge 2`
`:. sec^(-1) (x+1/x) in [pi/3, pi/2) uu (pi/2, (2pi)/3]`