cos(atan(sqrt(3))+asin(1/3)

cos( atan( sqrt(3) )+asin(1/3) ) ?

$$\boxed{ \mathbf{ \cos{ \left(~ \arctan{(~\sqrt{3}~)} + \arcsin{\left(\dfrac{1}{3}\right)} ~\right) } } =~ ? }$$

$$\small{\text{$ \begin{array}{rcl} \mathbf{ \cos{ \left(\alpha_{rad} + \beta_{rad} \right) } = \cos{ \left(\alpha_{rad} \right) } \cdot \cos{ \left(\beta_{rad} \right) } - \sin{ \left(\alpha_{rad} \right) } \cdot \sin{ \left(\beta_{rad} \right) } } & \quad \mathbf{ \alpha_{rad} = \arctan{(a)} \quad a= \sqrt{3} } \\ & \mathbf{ \quad \beta_{rad} = \arcsin{(b)} \quad b= \dfrac{1}{3} } \end{array} $}}$$

$$\small{\text{$ \begin{array}{rcl} \cos{ \left( \alpha_{rad}+ \beta_{rad} \right) } &=& \cos{ \left( \arctan{(a)} \right) } \cdot \cos{ \left(\arcsin{(b)} \right) } - \sin{ \left( \arctan{(a)} \right) } \cdot \sin{ \left(\arcsin{(b)} \right) } \\ &=& \cos{ \left( \arctan{(a)} \right) } \cdot \cos{ \left(\arcsin{(b)} \right) } - \sin{ \left( \arctan{(a)} \right) } \cdot b \end{array} $}}$$

$$\small{\text{ $ \boxed{ \cos{ \left(~ \arctan{(a)} ~\right) } =\pm\dfrac{1}{\sqrt{1+a^2}} =\pm\dfrac{1}{\sqrt{1+(\sqrt{3})^2}} =\pm \dfrac{1}{2} } $}}$$

$$\small{\text{ $ \boxed{ \cos{ \left(~ \arcsin{(b)} ~\right) } =\pm\sqrt{1-b^2} =\pm \sqrt{1+ \left( \dfrac{1}{3} \right)^2} =\pm \dfrac{\sqrt{8} }{ 3 } } $}}$$

$$\small{\text{ $ \boxed{ \sin{ \left(~ \arctan{(a)} ~\right) } =\pm\dfrac{a}{\sqrt{1+a^2}} =\pm\dfrac{ \sqrt{3} }{\sqrt{1+(\sqrt{3})^2}} =\pm \dfrac{ \sqrt{3} }{2} } $}}$$

$$\small{\text{$ \begin{array}{rcl} \cos{ \left( \alpha_{rad}+ \beta_{rad} \right) } &=& \left( \pm \dfrac{1}{2} \right) \cdot \left( \pm \dfrac{ \sqrt{8} }{ 3 } \right) -\left( \pm \dfrac{ \sqrt{3} }{ 2 } \right) \cdot \left( \dfrac{1}{3} \right) \\ &=& \dfrac{1}{6} \left( \pm \sqrt{8} \pm\sqrt{3} \right) \end{array} $}}$$

$$\small{\text{$ \begin{array}{rcl} \cos{ \left(~ \arctan{(~\sqrt{3}~)} + \arcsin{\left(\dfrac{1}{3}\right)} ~\right) } &=& \dfrac{1}{6} \left( + \sqrt{8} + \sqrt{3} \right) = 0.76007965539 \\\\ &=& \dfrac{1}{6} \left( + \sqrt{8} - \sqrt{3} \right) = 0.18272938620 \\\\ &=& \dfrac{1}{6} \left( - \sqrt{8} + \sqrt{3} \right) = -0.18272938620 \\\\ &=& \dfrac{1}{6} \left( - \sqrt{8} - \sqrt{3} \right) = -0.76007965539 \end{array} $}}$$

That looks very impressive Heureka :))