The solution set of `"sin" (x - (pi)/(4))-"cos" (x + (3 pi)/(4)) =1 " and " (2 "cos" 7x)/("cos" 3 + "sin" 3) gt 2^("cos" 2x)`, is

Options

A. `k pi + (-1)^(k) (pi)/(4), k in Z`
B. `(8k + 3) (pi)/(4), k in Z`
C. `(8k + 1) (pi)/(4), k in Z`
D. none of these

Answer

B

Explanation

We have,
`"sin" (x-(pi)/(4)) - "cos" (x + (3pi)/(4)) = 1`
`rArr "sin" x = (1)/(sqrt(2)) rArr x = 2n pi + (pi)/(4), 2n pi + (3pi)/(4), n in Z`
For these values of x, we have
`"cos" 2x = 0`
Also, `"cos" 3 + "sin" 3 lt 0`
`therefore (2"cos" 7x)/("cos"3 + "sin" 3) gt 2^("cos" 2x) rArr (2"cos" 7x)/("cos" 3 + "sin" 3) gt 1`
`rArr "cos" 7x lt 0`.
Clearly, `x = 2n pi + (3 pi)/(4), n in Z` satisfy this condition.