If the values of x between 0 and 2 `pi` which satisfy the equation `"sin" x|"cos"x| = (1)/(2sqrt(2))` are in A.P, then the common difference of the A.P, is

Options

A. `(pi)/(8)`
B. `(pi)/(4)`
C. `(3pi)/(8)`
D. `(5pi)/(8)`

Answer

B

Explanation

CASE I When`"cos" x ge 0 i.e. x in (0, pi//2) cup(3 pi//2, 2 pi)`
In this casae, we have
`|"cos"x|= "cos" x`
`therefore "sin"x |"cos"x| = (1)/(2sqrt(2))`
`rArr "sin" x "cos"x = (1)/(2sqrt(2))`
`rArr "sin" 2x = (1)/(sqrt(2)) rArr = 2x = (pi)/(4), (3pi)/(4) rArr x = (pi)/(8), (3pi)/(8)`
CASE II When `"cos" x lt 0 i.e. x in (pi//2, 3pi//2)`
In this case, we have
`|"cos"x | = -"cos"x`
`therefore "sin"x|"cos"x| = (1)/(2sqrt(2))`
`rArr -"sin"x"cos"x = (1)/(2sqrt(2))`
`rArr -"sin"2x = (1)/(sqrt(2))`
`rArr "sin" 2x = -(1)/(sqrt(2)) rArr 2x = (5pi)/(4),(7pi)/(4) rArr x = (5pi)/(8), (7pi)/(4)`
Thus, the values of x are : `(pi)/(8), (3pi)/(8), (5pi)/(8), (7pi)/(8)`
Clearly, these values are in A.P. with common difference `(pi)/(4)`