How can simplify this equation so that
$$\left\langle\sum _{n=0}^7c_n\cos\left(nt\right),\:\sum \:_{n=0}^7c_n\cos\left(nt\right)\:\right\rangle$$
becomes
$$\pi \left|c_0\right|^2+\frac{\pi \:}{2}\sum \:_{n=1}^7\left|c_n\right|^2$$
My attempt
$$\left\langle\sum _{n=0}^7c_n\cos\left(nt\right),\:\sum \:_{n=0}^7c_n\cos\left(nt\right)\:\right\rangle$$
$$=\sum \:_{n=0}^7\left(\sum \:\:_{n=0}^7\left|c_n\right|^2\langle \cos\left(nt\right),\:\cos\left(nt\right)\rangle\right)$$
$$=\sum \:_{n=0}^7\left(\sum \:\:_{n=0}^7\left|c_n\right|^2\int _{_0}^{\pi }\cos^2\left(nt\right)dt\:\right)$$
$$=\sum \:_{n=0}^7\left(\sum \:\:_{n=0}^7\left|c_n\right|^2\frac{\pi }{2}\right)$$
I don't know how to go on from here.