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I notice there are actually two standard forms of trignometric polynomials:

  1. $ c_0+ \sum_{k=1}^n \sum_{\alpha +\beta =k} c_{\alpha ,\beta}\sin^\alpha(x) \cos^\beta(x)$

  2. $ c_0+\sum_{k=1}^n\{a_k\sin(kx) +b_k\cos(kx)\}$

I know how to convert form 2 to 1. But generally how can I convert 1 to 2?

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You use the same identities backwards. For example, $\sin 3x=3 \cos^2 x \sin x-\sin^3x=3\sin x -4 \sin^3 x$, so $\sin^3 x=\frac 14(3\sin x-\sin 3x)$

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