(1) and (2) follow directly from the definitions of the series.
(3). The -th summand
(the term which refers to the power with exponent )
in the cosine series
(the coefficients referring to , odd, are )
where in the last step we have split up the index set into even and odd numbers.
The -th summand in the
and the -th summand in the Cauchy product of
Hence, both sides of the addition theorem coincide in the even case. For an odd index the left-hand side is . Since in the cosine series only even exponents occur, it follows that in the Cauchy product of the two cosine series only exponents of the form with even occur. Since in the sine series only odd exponents occur, it follows that in the Cauchy product of the two sine series only exponents of the form with even occur. Therefore terms of the form with odd occur neither on the left nor on the right-hand side.
The addition theorem for sine is proved in a similar way.
(4). From the addition theorem for cosine, applied to
and because of (2), we get