Question

If `sin^(- 1)(x-(x^2)/2+(x^3)/4-.....)+cos^(- 1)(x^2-(x^4)/2+(x^6)/4-.....)=pi/2` for `0 lt |x| lt sqrt2` then `x=`
A. `1//2`
B. `1`
C. `-1//2`
D. `-1 `

Answer 1

Correct Answer - B
We know that, ` sin ^(-1) ( alpha) + cos ^(-1) ( alpha ) = ( pi)/(2)`
Therefore, ` alpha ` should be equal in both functions.
`therefore x - ( x^(2))/( 2) + ( x ^(3))/(4) - … = x ^(2) - (x ^(4))/( 2) + (x ^(6))/( 4) - … `
`rArr (x)/( 1+ (x)/(2)) = ( x^(2))/( 1 + (x^(2))/( 2)) rArr (x)/(( 2+ x )/( 2)) = (x^(2))/(( 2+ x ^(2))/( 2 ))`
`rArr 2x ( 2 + x ^(2)) = 2x ^(2 ) (x + x )`
` rArr " " 4x + 2 x ^(3) = 4x ^(2) + 2x ^(3)`
`rArr " " x ( 4 + 2x ^(2) - 4x - 2x ^(2)) = 0`
`rArr ` Either ` x = 0 or 4- 4x = 0 `
`rArr x = 0 or x = 1 `
` because 0lt |x| lt sqrt2`
`therefore x= 1 and x ne 0`