Normally I just look up or remember that cos(π/4)=2√2cos(π/4)=22, or type "cos(π/4)cos(π/4)" into WolframAlpha to check the answer.
But what about the first time someone wanted to know what cos(π/4)cos(π/4) was? How could they find out before trig tables and WolframAlpha?
How can you prove that cos(π/4)=2√2cos(π/4)=22?