If `alpha and beta` are roots of the equatioin `a cos theta + b sin theta = c`, then find the value of `tan (alpha + beta).`

Explanation

Given equation is a `cos theta + b sin theta = c`
or `a + b tan theta = c sec theta` (dividing both sides by `cos theta`)
`rArr a^(2) + b^(2) tan^(2)theta + 2ab tan theta = c^(2) sec^(2) theta`
`rArr a^(2) + b^(2) tan^(2) theta + 2ab tan theta = c ^(2) (1 + tan^(2) theta)`
`rArr (b^(2) - c^(2)) tan^(2) theta + 2ab tan theta + a^(2) - c^(2) = 0`
Since roots of equation are `alpha and beta` , we have
`tan alpha + tan beta = - (2ab)/(b^(2) - c^(2))`
and `tan alpha tan beta = (a^(2) - c^(2))/(b^(2) -c^(2))`
`tan (alpha + beta) = (tanalpha + tan beta)/(1-tan alpha tan beta) = (-(2ab)/(b^(2) - c^(2)))/(1-(a^(2) - c^(2))/(b^(2) -c^(2))) = (2ab)/(a^(2) - b^(2))`