Sum of cosines of complementary/suplementary angles Why are $(\cos(2^{\circ})+\cos(178^{\circ})), (\cos(4^{\circ})+\cos(176^{\circ})),.., (\cos(44^{\circ})+\cos(46^{\circ}))$ all equal zero?
Could you prove it by some identity?

Three relevant identities:
$\cos(90^{\circ} - x) = \sin(x)$,
$\sin(90^{\circ} - x) = \cos(x)$,
$\sin(-x) = - \sin(x)$.
Therefore,
\begin{align*}
\cos(180^{\circ} - x) &= \cos(90^{\circ} - (x - 90^{\circ}))\\
 &= \sin(x - 90^\circ)\\
 &= -\sin(90^\circ - x)\\
 &= -\cos(x).
\end{align*}
