If `sin^(-1)sqrt(x^2+2x + 1) + sec^(-1)sqrt(x^2 + 2x + 1) = pi/2; x!= 0,` then the value of `2sec^(-1)(x/2) + sin^(-1)(x/2)` is equal to

Options

A. `-(pi)/(2)` only
B. `{-(3pi)/(2),(pi)/(2)}`
C. `(3pi)/(2)` only
D. `-(3pi)/(2)` only

Explanation

We have
`sin^(-1)sqrt(x^(2)+2x+1)+sec^(-1)sqrt(x^(2)+2x+1)=(pi)/(2)`
`sin^(-1)sqrt(x^(2)+2x+1)+cos^(-1)sqrt(1)/(x^(2)+2x+1)`
`rarr x^(2)+2x+1=1 rarr x=0 -2`
for x =0 we find that `sec^(-1)(-1)+sin^(-1)(-1)`
`=2xxpi-(pi)/(2)=(3pi)/(2)`