Would a straight line between two points on a sinusoidal map be the same as the geodesic between those points on a sphere's surface?

Question

Recently I've been trying to find the relationship between a Mercator map ( whose coordinates I'll express with x and y ) and the surface of a sphere such as a globe ( whose coordinates I'll express with λ and φ). I figured that $$[x, y] = [λcos(φ), φ]$$ I then calculated the metric tensor of this vector (g_λφ) and I noticed the g₁₁ = $(λ*sin(φ))^2 + 1$ At first I got confused because latitude distances should be equal on a Mercator and a globe, but then I realized that the term being influenced by longitude is a result of line distortion on sinusoidal maps - which means my original equation refers not to a sphere's surface, but a sinusoidal projection! Even so, I still feel like I'm onto something. Seeing as my original goal was to be able to calculate geodesics on a sphere and convert them to a Mercator projection, I would like to know if a geodesic on a sinusoidal is the same as that on a globe, because if so, I would have achieved this goal, and all this math would have not been a waste of time.