If `theta in (pi//4, pi//2)` and `sum_(n=1)^(oo)(1)/(tan^(n)theta)=sin theta + cos theta`, then the value of tan `theta` is

Options

A. `sqrt(3)`
B. `sqrt(2)+1`
C. `2+sqrt(3)`
D. `sqrt(2)`

Answer

A

Explanation

`theta in (pi//4, pi//2)`
`rArr tan theta gt 1 rArr 0 lt (1)/(tan theta)lt 1`
`sum_(n=1)^(oo)(1)/(tan^(n)theta)=sin theta + cos theta`
`rArr (1)/(tan theta)+(1)/(tan^(2)theta)+(1)/(tan^(3)theta)+….oo = sin theta + cos theta`
`rArr (1)/(tan theta -1)=sin theta + cos theta`
`rArr (cos theta)/(sin theta - cos theta)=sin theta + cos theta`
`rArr cos theta = sin^(2)theta-cos^(2)theta=1-2cos^(2)theta`
`rarr 2cos^(2)theta+cos theta-1=0`
`rArr (2cos theta -1)(cos theta +1)=0`
`rarr cos theta = (1)/(2)` or `cos theta = -1` (rejected)
`rarr theta =(pi)/(3)rArr tan theta = sqrt(3)`