Question

Prove 2 cos^2 B - cosB = 0

Answer 1

It cannot be proven for every value of B.
You can factor it:

cosB(2cosB - 1)=0.

If it is just an equation, it can be solved: it is true if cosB = 0 or if cosB = ½.
Solve these for B and you'll have done all you can ;)