Question

please prove the following Sum to product formulas:

*Sin a + sin b = 2sin1/2(a+b)cos1/2(a-b)
*Sin a - sin b = 2cos1/2(a+b)sin1/2(a-b)
*cos a + cos b = 2cos1/2(a+b)cos1/2(a-b)
*cos a - cos b = -2sin1/2(a+b)sin1/2(a-b)

advance thanks

Answer 1

Try from R.H.S

Answer 2

ill do the first one for example\[\sin \left( x+y \right)=\sin x \cos y + \cos y \sin x\]\[\sin \left( x-y \right) = \sin x \cos y - \cos y \sin x\]\[\sin \left( x+y \right) + \sin \left( x - y \right) = 2 \sin x \cos y\]Let\[a = x + y\]\[b = x - y\]\[\sin a + \sin b = 2 \sin \frac{1}{2}\left( a+b \right) \cos \frac{1}{2}\left( a - b \right)\]