Question

Which of the following is the solution set of the equation `2cos^(-1)x=cot^(-1)((2x^2-1)/(2xsqrt(1-x^2)))?` (0, 1) `(b)` (-1, 1) -{0} `(-1,0)` (d) `(-oo,-1)uu[1,oo]`
A. `(0,1)`
B. `(-1, 1) - {0}`
C. `(-1, 0)`
D. `[-1, 1]`

Answer 1

Correct Answer - A
`2 cos^(-1) x = cot^(-1) ((2x^(2) -1)/(2x sqrt(1 -x^(2))))`
Put `x = cos theta : L.H.S. = 2 theta, 0 le theta le pi`
and `-1 le x le 1`...(i)
`R.H.S. = cot^(-1) ((cos 2 theta)/(2 cos theta|sin theta|)) = cot^(-1) (cot 2 theta) = 2 theta`
if `0 lt 2 theta le pi`...(ii)
From Eqs. (i) and (ii) , we get `0 lt theta lt pi//2`. Thus,
`x in (0, 1)`