The number of roots of `(1-tan theta) (1+sin 2 theta)=1+tan theta` for `theta in [0, 2pi]` is

Options

A. 3
B. 4
C. 5
D. none of these

Answer

C

Explanation

`(1-tan theta) [1+2 tan theta//(1+tan^(2) theta)]=1+tan theta`
or `(1-tan theta) (1+tan theta)^(2) =(1+tan theta) (1+tan^(2) theta)`
or `(1+tan theta) [(1-tan^(2) theta)-(1+tan^(2) theta)]=0`
or `-2 tan^(2) theta=0, (1+tan theta)=0`
or `tan theta=0, or tan theta =-1`
`rArr theta n pi or n pi-pi//4, AA n in Z`.
For `theta in [0, 2pi], theta =0, pi, 2pi, 3pi//4, 7pi//4`