Day 14: Christmas is the Solution

Anti-solution: We notice that the title is "Christmas is the Solution". Being too lazy to actually solve the problem, we recall that Christmas is on 12/25. The question asks for a+b, so we assume that {a, b} = {12, 25}. Adding results in the correct answer of 37, by chance.

Good solution: Using $\cos^2{x}+\sin^2{x}=1$ , the fraction simplifies down to $\frac{\sin^2{x/12}}{\cos^2{x/12}}=\tan^2{x/12}$ . Using $\tan^2{x}+1=sec^2{x}$ , we are left with the integral of $\sec^2{x/12}$ , which evaluates to $12\tan{x/12}$ . Evaluating from 0 to $3\pi$ results in the value 12. Substituting into the given format gives the fraction $\frac{25}{12}$ , so $a+b=37$ .

A bit of simplifying shows that $C$ is the integral of $\sec^2{\frac{x}{12}}$ which gives: $\left[ 12\tan{\frac{x}{12}} \right]_0^{3\pi} = 12$ Therefore $2 + \frac{12}{144} = 25/12$ and so the final answer is $\boxed{37}$ .