If `x+y=pi//4` and `tan x+tan y=1`, then `(n in Z)`

Options

A. `x = (pi)/(2)-n pi "and " y = n pi, n in Z`
B. `x = (pi)/(4) - n pi "and " y = n pi, n in Z`
C. `x = (pi)/(4)-n pi " and " y = 2n pi, n in Z`
D. none of these

Answer

B

Explanation

We have,
`x + y = (pi)/(4) " "..(i)` ltbr `"and, tan"x + "tan" y = 1" "…(ii)`
Now,
`"tan" (x+y) = ("tan"x + "tan"y)/(1-"tan"x "tan" y)`
`rArr "tan" x "tan" y = 0" "...(iii)`
From (ii) and (iii), we have
`("tan" x = 0 "and tan" y = 1)"or" ("tan"x =1 "and tan"y = 0)`
`rArr (x = n pi "and " y = -n pi + (pi)/(4), (n in Z)`
`"or" (x = (pi)/(4)-n pi "and"y = n pi, n in Z)`