Proofs

Trigonometric Proof

Prove sinu + sinv = 2sin (u+v/2) cos (u-v/2)

1. 2sin (u+v/2) cos (u-v/2)	1. Given
2. sinu cosv = ½ (sin (u+v) + sin (u-v))	2. This is the product-to-sum formula, ½ (sin (u+v) + sin (u-v)) that was also given.
3. 2 x ½ (sin (u+v – (u-v)/s) + sin (u+v+u-v)/2)	3. Plug in the equation 2sin (u+v/2) cos (u-v/2)
4. 1 (sin (2v/2) + sin (2u/2))	4. Distributive property of multiplication and combine like terms.
5. 1 (sinv + sinu) = sinv + sinu	5. Simplify!

Paragraph Proofs: I would like to prove that sinu + sinv = 2sin (u+v/2) cos (u-v/2). I was given that sinu cosv = ½ (sin (u+v) + sin (u-v)) so I plugged 2sin (u+v/2) cos (u-v/2) into ½ (sin (u+v) + sin (u-v)) and I got 2 x ½ (sin (u+v – (u-v)/s) + sin (u+v+u-v)/2). Then I used the distributive property of multiplication and combined like terms to get 1 (sin (2v/2) + sin (2u/2)) and lastly, I simplified the equation to get sinv + sinu.