If `alpha` and `beta` are solutions of `sin^2 x + a sin x+b=0` as well as that of `cos^2x + c cosx + d =0` then `sin(alpha + beta) ` is equal to

Options

A. `(2bd)/(b^(2)+d^(2))`
B. `(a^(2)+c^(2))/(2ac)`
C. `(b^(2)+d^(2))/(2bd)`
D. `(2ac)/(a^(2)+c^(2))`

Answer

D

Explanation

If `alpha` and `beta` are …………..
According to the given condition, `sin alpha + sin beta = -a` and `cos alpha+cos beta= -c` .
`implies 2 sin. (alpha+beta)/(2)cos.(alpha-beta)/(2)= -a &`
`2cos.(alpha+beta)/(2)cos.(alpha-beta)/(2)= -c`
`implies tan. (alpha+beta)/(2)= (a)/(c ) implies sin. (alpha + beta)`
`(2tan.(alpha+beta)/(2))/(1+tan.^(2)(alpha+beta)/(2)) = (2ac)/(a^(2)+c^(2))`