19. The graph of function $f ( x )$ is obtained from the graph of function $g ( x ) = \cos x$ by the following transformations: first, stretch the vertical coordinates of all points on the graph of $g ( x )$ to 2 times their original length (horizontal coordinates unchanged), then shift the resulting graph to the right by $\frac { \pi } { 2 }$ units. (1) Find the analytical expression of function $f ( x )$ and the equation of its axis of symmetry; (2) Given that the equation $f ( x ) + g ( x ) = m$ has two distinct solutions $a$ and $b$ in $[ 0,2 \pi )$: 1) Find the range of the real number $m$; 2) Prove that $\cos ( a - b ) = \frac { 2 m ^ { 2 } } { 5 } - 1$.
19. The graph of function $f ( x )$ is obtained from the graph of function $g ( x ) = \cos x$ by the following transformations: first, stretch the vertical coordinates of all points on the graph of $g ( x )$ to 2 times their original length (horizontal coordinates unchanged), then shift the resulting graph to the right by $\frac { \pi } { 2 }$ units.\\
(1) Find the analytical expression of function $f ( x )$ and the equation of its axis of symmetry;\\
(2) Given that the equation $f ( x ) + g ( x ) = m$ has two distinct solutions $a$ and $b$ in $[ 0,2 \pi )$:\\
1) Find the range of the real number $m$;\\
2) Prove that $\cos ( a - b ) = \frac { 2 m ^ { 2 } } { 5 } - 1$.