Let `alpha, beta` are the root of equation `acostheta + bsintheta = c`.
Which of the following is/are true.

Options

A. `sinalpha + sinbeta= (2bc)/(a^(2) + b^(2))`
B. `sinalpha + sinbeta= (2bc)/(b^(2) + c^(2))`
C. `tan'(alpha)/(2) + tan'(beta)/(2) = (b)/(a + c)`
D. `tan'(alpha)/(2) + tan"(beta)/(2) = (2b)/(a + c)`

Answer

A, D

Explanation

`"Sum" = sinalpha + sinbeta = (2bc)/(a^(2) + b^(2))`
Product `= sinalpha sinbeta = (c^(2) - a^(2))/(c^(2) - a^(2))`
`tan ((alpha)/(2)) + tan ((beta)/(2)) = (2b)/(a + c)`
bacause equation is `(a^(2) - b^(2))sin^(2)theta - 2bc sintheta + c^(2) - a^(2) = 0`
or `(a + c)tan^(2)((theta)/(2)) - 2btan((theta)/(2)) + c - a = 0`