Find the number of roots of the equation `16sec^3theta-12tan^2theta-4sectheta=9` in interval `(-pi,pi)`

Explanation

We have `16 sec^(3) theta-12 (sec^(2) theta-1) -4` sec `theta-9=0`
`rArr 16 sec^(3) theta-12 sec^(2) theta-4` sec `theta+3=0`
Let sec `theta=t`
`rArr 16t^(3)-12t^(2)-4t+3=0`
`rArr (2t-1) (2t+1) (4t-3)=0`
`:. T=1/2, (-1)/2, 3/4`
`rArr cos theta=2, -2, 4/3` (Not possible)
`:.` Thus, equation has no real roots.