If roots of equation `3x^(2) + 5x + 1 = 0` are `(sectheta_(1) - tentheta_(1)) and (cosectheta_(2)-cottheta_(2)),` then find the equation whose roots are `(sectheta_(1)+tantheta_(1) and (cosectheta_(2) + cottheta_(2)).`

Explanation

Let `sectheta_(1) - tantheta_(1) = alpha then sectheta_(1) + tantheta_(1) = (1)/(alpha)`
and `cosectheta_(2) - cottheta_(2) = beta, then cosectheta_(2) + cottheta_(2) = (1)/(beta)`
Hence, we have to find the equation having roots `1//alpha and 1/beta`
Replacing x by `1/x` in the given equation, we get required equation as
`3((1)/(2))^(2) + 5 ((1)/(x)) + 1 = 0`
or ` x^(2) + 5x + 3 = 0 `