What is the Simplified Form of This Trigonometric Identity?

[Silver Bolt]

$\sin\left({A}\right)+\sin\left({A+\frac{2\pi}{3}}\right)+\sin\left({A+\frac{4\pi}{3}}\right)=0$

$L.H.S=\sin\left({A}\right)+\left(\sin\left({A}\right)\cos\left({\frac{2\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{2\pi}{3}}\right)\right)+\left(\sin\left({A}\right)\cos\left({\frac{4\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{4\pi}{3}}\right)\right) $

From there?

 

[MarkFL]

You've only given an expression...what is the actual identity to be verified?

 

[Silver Bolt]

Corrected now

 

[MarkFL]

$\sin\left({A}\right)+\sin\left({A+\frac{2\pi}{3}}\right)+\sin\left({A+\frac{4\pi}{3}}\right)=0$

$L.H.S=\sin\left({A}\right)+\left(\sin\left({A}\right)\cos\left({\frac{2\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{2\pi}{3}}\right)\right)+\left(\sin\left({A}\right)\cos\left({\frac{4\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{4\pi}{3}}\right)\right) $

From there?

I would write the LHS as:

\(\displaystyle \sin(A)\left(1+\cos\left(\frac{2\pi}{3}\right)+\cos\left(\frac{4\pi}{3}\right)\right)+\cos(A)\left(\sin\left(\frac{2\pi}{3}\right)+\sin\left(\frac{4\pi}{3}\right)\right)\)

Now, what are:

\(\displaystyle 1+\cos\left(\frac{2\pi}{3}\right)+\cos\left(\frac{4\pi}{3}\right)=?\)

\(\displaystyle \sin\left(\frac{2\pi}{3}\right)+\sin\left(\frac{4\pi}{3}\right)=?\)