I have the expression $$1=\sum^\infty_{n=1} c_n \left(\frac{\pi n}{3}\right) \cosh \left(\frac{2 \pi n y}{3}\right) \sin\left(\frac{\pi n y}{3}\right)$$
I am trying to find the constant $$c_n$$
The first thing I did was to multiply both sides by $$\sin\left(\frac{\pi m y}{3}\right)$$ I also let $$m=n$$ This gave me $$sin\left(\frac{\pi n y}{3}\right)=\sum^\infty_{n=1} c_n \left(\frac{\pi n}{3}\right) \cosh\left(\frac{2 \pi n y}{3}\right) \sin^2 \left(\frac{\pi n y}{3}\right)$$
I look at the RHS: $$\int^3_0 \left(\frac{\pi n }{3}\right) c_n \cosh \left(\frac{2 \pi n y}{3}\right) \sin^2\left(\frac{\pi n y}{3}\right)dy$$
I solve this then make the answer equal to $$\int^3_0 \sin\left(\frac{\pi n y}{3}\right) dy$$ and then solve for $$c_n$$
I followed this method but I got a completely wrong answer and I am not sure where?