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

Options

A. `cos(alpha-beta)`
B. `sin(alpha-beta)`
C. `cos(alpha+beta)`
D. none of these.

Answer

A

Explanation

`(a)/(sin(theta-alpha))=(b)/(sin(theta-beta))=k_(1)`
`(c)/(cos(theta-alpha))=(d)/(cos(theta-beta))=k_(2)`
`sin(theta-alpha)cos (theta-alpha)`
`rArr(ac+bd)/(ad+bc)=(k_(1)k_(2))/(k_(1)k_(2))(+sin(theta-beta)cos(theta-beta))/(sin(theta-alpha)cos(theta-beta))`
`+sin(theta-beta)cos(theta-alpha)`
`=(sin2(theta-alpha)+sin2(theta-beta))/(2sin(2theta-(alpha+beta)))`
`(2sin(2theta-(alpha+beta))cos (alpha-beta))/(2sin(2theta-(alpha+beta)))=cos (alpha-beta)`