Prove that $\sin a + \sin b + \sin(a+b) = 4 \sin\frac12(a+b) \cos \frac12a \cos\frac12b$ Helping my son with his trigonometry review. We know that 
$$\sin(a+b) = \sin a \cos b + \cos a \sin b$$ 
We also know that 
$$\sin a \cos b = \frac12 \left(\sin(a-b) + \sin(a+b)\right)$$
And we have 
$$\sin a + \sin b = 2 \sin\frac12(a+b)\cos\frac12(a-b)$$
From there, we seem to be missing how to get to the right-hand side of the equation.
We first expand 
$$\sin a + \sin b = 2 \sin\frac12(a+b) \cos\frac12(a-b)$$
Then we add $\sin(a+b)$, which is $\sin a \cos b + \cos a \sin b$.
We now have:
$$2 \sin\frac12(a+b) \cos\frac12(a-b) + \frac12 \left(\sin(a-b) + \sin (a+b)\right) + \frac12 \left( \sin(b-a) + \sin (a+b)\right)$$
From there, we can't see how to obtain the right-hand side of the equation which is 
$$4 \sin\frac12(a+b) \cos\frac12a \cos\frac12b$$ 
 A: Following Thomas Shelby and Dr. Sonnhard Graubner, I provide you a full proof:
\begin{align}\sin(a)+\sin(b)+\sin(a+b)&=2\cos\left(\frac{a-b}{2}\right)\sin\left(\frac{a+b}{2}\right)+2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a+b}{2}\right)\\&=2\sin\left(\frac{a+b}{2}\right)\left[\cos\left(\frac{a-b}{2}\right)+\cos\left(\frac{a+b}{2}\right)\right]\end{align}
Since 
\begin{align}\cos\left(a-b\right)+\cos\left(a+b\right)=2\cos(a)\cos(b)\end{align}
[see: last page of https://services.math.duke.edu/~leili/teaching/uwm/math222s11/problems/quizzes/trig.pdf or search for An elementary proof of two formulas in trigonometry on google] then,
\begin{align}\cos\left(\frac{a-b}{2}\right)+\cos\left(\frac{a+b}{2}\right)=2\cos\left(\frac{a}{2}\right)\cos\left(\frac{b}{2}\right)\end{align}
Hence, 
\begin{align}2\sin\left(\frac{a+b}{2}\right)\left[\cos\left(\frac{a-b}{2}\right)+\cos\left(\frac{a+b}{2}\right)\right]=&2\sin\left(\frac{a+b}{2}\right)\times 2\cos\left(\frac{a}{2}\right)\cos\left(\frac{b}{2}\right)\\=&4\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a}{2}\right)\cos\left(\frac{b}{2}\right)\end{align}
A: Use that $$\sin(A)+\sin(B)=2\cos\left(\frac{A-B}{2}\right)\sin\left(\frac{A+B}{2}\right)$$ and $$\sin(A+B)=2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A+B}{2}\right)$$
A: It may help setting $A=a/2$ and $B=b/2$, so the left-hand side becomes
$$
\sin2A+\sin2B+\sin2(A+B)
$$
The right-hand side screams “sum-to-product”! OK, let's apply the formula
$$
\sin2A+\sin2B=2\sin(A+B)\cos(A-B)
$$
so the left-hand side becomes
\begin{align}
\sin2A+\sin2B+\sin2(A+B)
&=2\sin(A+B)\cos(A-B)+2\sin(A+B)\cos(A+B)\\
&=2\sin(A+B)\bigl(\cos(A-B)+\cos(A+B)\bigr) \\
&=2\sin(A+B)(\cos A\cos B+\sin A\sin B+\cos A\cos B-\sin A\sin B)\\
&=4\sin(A+B)\cos A\cos B
\end{align}
A: \begin{align}
\sin a + \sin b + \sin (a+b)&=2 \sin\left(\frac{a+b}2\right) \cos\left(\frac{a-b}2\right)+2 \sin\left(\frac{a+b}2\right) \cos\left(\frac{a+b}2\right)\\
&=2 \sin\left(\frac{a+b}2\right)\left(\cos\left(\frac{a-b}2\right)+\cos\left(\frac{a+b}2\right)\right)
\end{align}
Now use the identity  $$\cos A +\cos B=2\cos \left(\frac{A+B}2\right)\cos \left(\frac{A-B}2\right)$$.
A: An alternative strategy uses $1+\cos x=2\cos^2\tfrac{x}{2}$ to obtain $$\sin a+\sin b+\sin (a+b)=2\left(\sin a\cos^2\tfrac{b}{2}+\sin b\cos^2\tfrac{a}{2}\right).$$To finish, use $\sin x=2\sin\tfrac{x}{2}\cos\tfrac{x}{2}$ to obtain$$\sin a+\sin b+\sin (a+b)=4\cos\tfrac{a}{2}\cos\tfrac{b}{2}\left(\sin\tfrac{a}{2}\cos\tfrac{b}{2}+\sin\tfrac{b}{2}\cos\tfrac{a}{2}\right)=4\cos\tfrac{a}{2} \cos\tfrac{b}{2}\sin\tfrac{a+b}{2}.$$
