\[\boxed{\mathbf{506}\mathbf{.}}\]
\[1)\ 2\cos{40{^\circ}} \bullet \cos{50{^\circ}} =\]
\[= 2 \bullet \cos(90{^\circ} - 50{^\circ}) \bullet \cos{50{^\circ}} =\]
\[= 2\sin{50{^\circ}} \bullet \cos{50{^\circ}} =\]
\[= \sin(2 \bullet 50{^\circ}) = \sin{100{^\circ}} =\]
\[= \sin{80{^\circ}}\]
\[2)\ 2\sin{25{^\circ}} \bullet \sin{65{^\circ}} =\]
\[= 2 \bullet \cos(90{^\circ} - 65{^\circ}) \bullet \sin{65{^\circ}} =\]
\[= 2\cos{65{^\circ}} \bullet \sin{65{^\circ}} =\]
\[= \sin(2 \bullet 65{^\circ}) = \sin{130{^\circ}} =\]
\[= \sin{50{^\circ}}\]
\[3)\sin{2a} + \left( \sin a - \cos a \right)^{2} =\]
\[= \sin{2a} + 1 - \sin{2a} = 1\]
\[4)\cos{4a} + \sin^{2}{2a} =\]
\[= \cos{4a} + \sin^{2}{2a} =\]
\[= \cos^{2}\frac{4a}{2} - \sin^{2}\frac{4a}{2} + \sin^{2}{2a} =\]
\[= \cos^{2}{2a} - \sin^{2}{2a} + \sin^{2}\text{ax} =\]
\[= \cos^{2}{2a}\]