Question

If sin A + sin B = C, cos A + cos B = D, then the value of sin(A + B) = ______.

Options

• CD

• `(CD)/(C^2 + D^2)`

• `(C^2 + D^2)/(2CD)`

• `(2CD)/(C^2 + D^2)`

Answer 1

If sin A + sin B = C, cos A + cos B = D, then the value of sin(A + B) = `underlinebb((2CD)/(C^2 + D^2))`.

Explanation:

Given: 

sin A + sin B = C 

cos⁡ A + cos⁡ B = D 

Dividing both sides: 

`(sin A + sin B)/(cos A + cos B) = C/D`

Using sum-to-product identities:

`(2 sin  (A + B)/2 cos  (A - B)/2)/(2 cos  (A + B)/2 cos  (A - B)/2` = `C/D`

`tan  (A + B)/2 = C/D`

`sin (A + B) = (2 C/D)/(1 + (C/D)^2)`

= `(2C/D)/(1 + C^2/D^2)`

sin (A + B) = `(2CD)/(C^2 + D^2)`