Deriving the length of action formula for involute gears

[A330NEO]

Homework Statement

Length of action Z(Length of line of action) is $Z=\sqrt{(r_{p}+a_{p})^{2}-(r_{p}cos\phi )^{2}}+\sqrt{(r_{g}+a_{g})^{2}-(r_{g}cos\phi )^{2}}-Ccos\phi$ How can I drive it?

Homework Equations

The Attempt at a Solution

 

[Nidum]

http://nptel.ac.in/courses/IIT-MADRAS/Machine_Design_II/pdf/2_2.pdf

Item 2.7