Step 1: Place the two forces symmetrically about a common axis.
Instead of the cosine rule, place both forces of magnitude \( P \) symmetrically, one at angle \( +\frac{\alpha}{2} \) and the other at angle \( -\frac{\alpha}{2} \) from a central bisecting line, so the angle between them is still \( \alpha \).
Step 2: Add the components along the bisector.
Along the bisector direction (call it the x-axis), both forces contribute:
\[
R_x = P\cos\left(\frac{\alpha}{2}\right) + P\cos\left(\frac{\alpha}{2}\right) = 2P\cos\left(\frac{\alpha}{2}\right)
\]
Step 3: Add the components perpendicular to the bisector.
Perpendicular to the bisector, the two forces point in opposite directions and cancel exactly:
\[
R_y = P\sin\left(\frac{\alpha}{2}\right) - P\sin\left(\frac{\alpha}{2}\right) = 0
\]
Since the resultant has no perpendicular component, its full magnitude is just the bisector component:
\[
R = \sqrt{R_x^2 + R_y^2} = R_x = \boxed{2P\cos\left(\frac{\alpha}{2}\right)}
\]
which matches option (C). This also shows directly why the resultant always points exactly along the bisector line of the two equal forces.