The number of solution of `sec^(2) theta + cosec^(2) theta+2 cosec^(2) theta=8, 0 le theta le pi//2` is

Options

A. 4
B. 3
C. 0
D. 2

Answer

D

Explanation

We have
`1/(sin^(2) theta cos^(2) theta)+2/(sin^(2) theta)=8, sin theta ne 0, cos theta ne 0`
or `1+2 cos^(2) theta =8 sin^(2) theta cos^(2) theta`
or `1+2 cos^(2) theta =8 cos^(2) theta (1- cos^(2) theta)`
or `8 cos^(4) theta-6 cos^(2) theta +1=0`
or `(4 cos^(2) theta -1) (2 cos^(2) theta -1)=0`
or `cos^(@) theta = 1//4= cos^(2) (pi//3)`
or `cos^(2) theta=1//2=cos^(2) (pi//4)`
or `theta=n pi pm (pi//3) or theta =npi pm (pi//4), n in Z`
Hence, for `0 le theta le pi/2, theta = pi//3, theta =pi//4`