Fun for nerds stuck in boring meetings

Most meetings I attend are %90 boring and only 10% useful. I found that if I have pencil and paper I can prevent boredom by doing some simple derivations. By "simple" I mean that the derivation should take less than 5 minutes so that I can still follow the meeting. Here are some examples that I use to keep myself busy:

• Show that θ = 1/2*arcsin(g*x/v^2) in projectile motion
• Show that θ is also the solution of g*x^2/(2*v^2)*tan(θ)^2 - x*tan(θ) + g*x^2/(2*V^2) = 0. Hint: Show that 1/cos(θ)^2 = 1+tan(θ)^2
• Show that arctan(x) = x - x^3/3 + x^5/5 - x^7/7 +...
• Show that sqrt(2) is irrational.
• Show that sin(θ)^2+cos(θ)^2 = 1
• Show that 0.999... = 1
• Show that for a right triangle c^2 = a^2+b^2