If `sin^(3) theta+sin theta cos theta+ cos^(3) theta=1`, then `theta` is equal to `(n in Z)`

Options

A. `2 n pi`
B. `2n pi +pi/2`
C. `2npi-pi/2`
D. `npi`

Answer

A, B

Explanation

`(sin^(3) theta+cos^(3)theta)-(1-sin theta cos theta)=0`
or `(sin theta + cos theta)(1-sin theta cos theta)-(1-sin theta cos theta)=0`
or `(1-sin theta cos theta) (sin theta + cos theta-1)=0`
If `sin theta cos theta=1`
`rArr 2 sin theta cos theta =2`
`rArr sin 2 theta=2` (not possible)
`rArr sin theta + cos theta=1`
`rArr cos (theta-pi/4)=1/sqrt(2)`
`rArr theta-pi/4=2npi pm pi/4, n in Z`
`rArr theta=2npi` or `2npi+pi/2`