Sin cos tan

Alpha + beta = 270 , cos alpha + sin beta = 0

$\small{ \begin{array}{rcll} \alpha + \beta &=& 270 \\ \beta &=& 270 - \alpha \\\\ \cos{(\alpha)} + \sin{(\beta)} &=& \cos{(\alpha)} +\sin{(270 - \alpha)} \\ && \sin{(270 - \alpha)} = \sin{(270 )}\cos{(\alpha)} + \cos{(270 )}\sin{(\alpha)} \\ &=& \cos{(\alpha)} +\sin{(270 )}\cos{(\alpha)} + \cos{(270 )}\sin{(\alpha)} \quad & | \quad \sin{(270 )} = -1 \qquad \cos{(270 )} = 0 \\ &=& \cos{(\alpha)} +(-1)\cdot \cos{(\alpha)} + 0\cdot \sin{(\alpha)}\\ &=& \cos{(\alpha)} -\cdot \cos{(\alpha)} \\ &=& 0 \\ \cos{(\alpha)} + \sin{(\beta)} &=& 0 \end{array} }$