[tex]a) \\ \\ $Consideram:$ \\-------------------------------\\ a = arccos(x)\\ \Rightarrow cos(a) = x \\ \\ b = arcsin(x) \\ \Rightarrow sin(b) = x \\------------------------------- \\ \\ \Rightarrow cos(a) = sin(b) \Leftrightarrow cos(a) = cos\Big( \dfrac{\pi}{2}-b\Big) \Rightarrow a = \dfrac{\pi}{2} - b \\ \\ $Inlocuim notatiile: $ \\ \\ \Rightarrow arccos(x) = \dfrac{\pi}{2} - arcsin(x) \Rightarrow \boxed{arccos(x) + arcsin(x) = \dfrac{\pi}{2} }[/tex]
[tex]b)[/tex]
[tex]arccos(x) - arcsin(x) = \dfrac{\pi}{6} \\ \\ arccos(x)+arcsin(x)=\dfrac{\pi}{2}\quad (identitate) \\ ------------- -$ $ ($adunam$) \\ \\ 2\cdot arccos(x) \quad \backslash \quad $ $ $ $ $ $ \quad = \dfrac{\pi}{6}+ \dfrac{\pi}{2} \\ \\ \Rightattow 2\cdot arccos(x) = \dfrac{4\pi}{6} \Rightarrow 2\cdot arccos(x) = \dfrac{2\pi}{3} \Rightarrow arccos(x) = \dfrac{\pi}{3} \Rightarrow \\ \\ \Rightarrow \boxed{x = \dfrac{1}{2}}[/tex]
