Let $\map f x$ be the function defined as: :$\forall x \in \openint 0 \pi: \map f x = \cos x$ Let $f$ be expressed by a half-range Fourier sine series: :$\displaystyle \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \cos x \sin \frac {n \pi x} \lambda \rd x$ In this context, $\lambda = \pi$ and so this can be expressed more simply as: :$\displaystyle \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \pi \int_0^\pi \cos x \sin n x \rd x$ First the case when $n = 1$: {{begin-eqn}} {{eqn | l = b_1       | r = \frac 2 \pi \int_0^\pi \cos x \sin x \rd x       | c =  }} {{eqn | r = \frac 2 \pi \intlimits {\frac {\sin^2 x} 2} 0 \pi       | c = Primitive of $\cos x \sin x$ }} {{eqn | r = \frac 2 \pi \paren {\frac {\sin^2 \pi} 2 - \frac {\sin^2 0} 2}       | c =  }} {{eqn | r = \frac 2 \pi \paren {0 - 0}       | c = Sine of Multiple of Pi }} {{eqn | r = 0       | c =  }} {{end-eqn}} When $n \ne 1$: {{begin-eqn}} {{eqn | l = b_n       | r = \frac 2 \pi \int_0^\pi \cos x \sin n x \rd x       | c =  }} {{eqn | r = \frac 2 \pi \intlimits {\frac {-\cos \paren {n - 1} x} {2 \paren {n - 1} } - \frac {\cos \paren {n + 1} x} {2 \paren {n + 1} } } 0 \pi       | c = Primitive of $\cos x \sin n x$ }} {{eqn | r = \frac 2 \pi \paren {\paren {\frac {-\cos \paren {n - 1} \pi} {2 \paren {n - 1} } - \frac {\cos \paren {n + 1} \pi} {2 \paren {n + 1} } } - \paren {\frac {-\cos 0} {2 \paren {n - 1} } - \frac {\cos 0} {2 \paren {n + 1} } } }       | c =  }} {{eqn | r = \frac 2 \pi \paren {\paren {\frac {-\cos \paren {n - 1} \pi} {2 \paren {n - 1} } - \frac {\cos \paren {n + 1} \pi} {2 \paren {n + 1} } } + \frac 1 {2 \paren {n - 1} } + \frac 1 {2 \paren {n + 1} } }       | c = Cosine of Zero is One and simplifying }} {{eqn | r = \frac 1 \pi \paren {\frac {1 - \cos \paren {n - 1} \pi} {\paren {n - 1} } + \frac {1 - \cos \paren {n + 1} \pi} {\paren {n + 1} } }       | c = Further simplification }} {{end-eqn}} Thus for $n = 2 m$ for $m \in \Z$: {{begin-eqn}} {{eqn | l = b_n       | r = \frac 1 \pi \paren {\frac {1 - \cos \paren {2 m - 1} \pi} {\paren {2 m - 1} } + \frac {1 - \cos \paren {2 m + 1} \pi} {\paren {2 m + 1} } }       | c =  }} {{eqn | r = \frac 1 \pi \paren {\frac {1 - \paren {-1} } {\paren {2 m - 1} } + \frac {1 - \paren {-1} } {\paren {2 m + 1} } }       | c = Cosine of Multiple of Pi }} {{eqn | r = \frac 1 \pi \paren {\frac 2 {\paren {2 m - 1} } + \frac 2 {\paren {2 m + 1} } }       | c =  }} {{eqn | r = \frac 1 \pi \paren {\frac {2 \paren {2 m + 1} + 2 \paren {2 m - 1} } {\paren {2 m + 1} \paren {2 m - 1} } }       | c =  }} {{eqn | r = \frac {4 m + 2 + 4 m - 2} {\pi \paren {2 m + 1} \paren {2 m - 1} }       | c =  }} {{eqn | r = \frac {8 m} {\pi \paren {4 m^2 - 1} }       | c =  }} {{end-eqn}} and for $n = 2 m + 1$ for $m \in \Z$: {{begin-eqn}} {{eqn | l = b_n       | r = \frac 1 \pi \paren {\frac {1 - \cos \paren {2 m} \pi} {\paren {2 m - 1} } + \frac {1 - \cos \paren {2 m + 2} \pi} {\paren {2 m + 1} } }       | c =  }} {{eqn | r = \frac 1 \pi \paren {\frac {1 - 1} {\paren {2 m - 1} } + \frac {1 - 1} {\paren {2 m + 1} } }       | c = Cosine of Multiple of Pi }} {{eqn | r = 0       | c =  }} {{end-eqn}} Thus we have: {{begin-eqn}} {{eqn | l = b_{2 m}       | r = \frac {8 m} {\pi \paren {4 m^2 - 1} } }} {{eqn | l = b_{2 m + 1}       | r = 0 }} {{end-eqn}} and so over the given interval: {{begin-eqn}} {{eqn | l = \cos x       | r = \sum_{m \mathop = 1}^\infty \frac {8 m} {\pi \paren {4 m^2 - 1} } \sin 2 m x }} {{eqn | l =        | r = \frac 8 \pi \sum_{m \mathop = 1}^\infty \frac {m \sin 2 m x} {\paren {4 m^2 - 1} } }} {{end-eqn}} {{qed}}