Let `(sin(theta-alpha))/("sin"(theta-beta))=a/b a n d(cos(theta-alpha))/("cos"(theta-beta))=c /d t h e n(a c+b d)/(a d+b c)=` (a)`cos(alpha-beta)` (b) `sin(alpha-beta)` (c)`sin(alpha+beta)` (d) none of these

Explanation

`sin(theta-alpha)/(sin(theta-beta)) = a/b`

`=>a/sin(theta-alpha) = b/(sin(theta-beta)) `

Let `a/sin(theta-alpha) = b/(sin(theta-beta)) = k_1`

`=>a = k_1sin(theta-alpha), b = k_1sin(theta-beta)`

Similarly,

`=>c = k_2cos(theta-alpha), d = k_2cos(theta-beta)`

Now,

`(ac+bd)/(ad+bc) = (k_1sin(theta-alpha)k_2cos(theta-alpha)+k_1sin(theta-beta)k_2cos(theta-beta))/(k_1sin(theta-alpha)k_2cos(theta-beta)+k_1sin(theta-beta)k_2cos(theta-alpha))`

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