+129845 arctan (1/4) = 14.04°
arccos (1/2) = 60°
So
cos ( arctan (1/4) + arccos (1/2) ) =
cos (14.04 + 60) =
cos (74.04) ≈ 0.275
Find the third side of the right-angled triangles and read off the other two ratios,
Angle A = \(\tan^{-1}(1/4)=\cos^{-1}(4/\sqrt{17})=\sin^{-1}(1/\sqrt{17}),\)
Angle B = \(\cos^{-1}(1/2)=\sin^{-1}(\sqrt{3}/2)=\tan^{-1}\sqrt{3}\).
\(\displaystyle \cos(A+B)= \cos(A)\cos(B)-\sin(A)\sin(B)\\=\frac{4}{\sqrt{17}}.\frac{1}{2}-\frac{1}{\sqrt{17}}.\frac{\sqrt{3}}{2}=\frac{4-\sqrt{3}}{2\sqrt{17}}\approx0.275029.\)
