The most general of `theta` satisfying `tantheta+tan((3pi)/(4)+theta)=2` are given by

Options

A. `2npi+-(pi)/(3),n in Z`
B. `n pi+(pi)/(3), n in Z`
C. `2n pi+-(pi)/(6),n in Z`
D. `n pi+-(pi)/(6), n in Z`

Answer

B

Explanation

`tantheta+tan((3pi)/(4)+theta)=2`
`impliestantheta+tan[(pi)/(2)+((pi)/(4)+theta)]=2`
`impliestantheta-cot((pi)/(4)-theta)=2`
`impliestantheta-("cot"(pi)/(4)cot theta-1)/("cot"(pi)/(4)+cot theta)=2`
`impliestan theta-(cot theta-1)/(1+cot theta)=2`
`impliestan theta-(1-tantheta)/(1+tan theta)=2`
`impliestan theta+tan^(2)theta-1+tan theta`
`=2+2tan theta`
`impliestan^(2)theta=3`
`impliestantheta=+-sqrt(3)=+-"tan"(pi)/(3)`
`impliestheta=n pi+-(pi)/(3),n inZ`