My younger son is studying in Art of Problem Solving’s Precalculus book this year. Right now he’s looking at some of the trig problems in

(1) The first problem asks you to prove that the area of a triangle is A*B*C / 4R, where A, B, and C are the side lengths and R is the radius of the circumscribed circle:

(2) The second problem asks you to prove that in an acute angled triangle that:

b = c Cos(A) + a Cos(C), where a, b, and c are the side lengths of the triangle and A and C are the angles opposite sides a and c.

(3) The third problem is Tan(A/2) = r / (S – A), where A is the angle opposite side A, r is the radius of the inscribed circle, and S is half the perimeter of the triangle.

(4) The final problem is pretty difficult -> you are asked to prove this identity:

It takes 10 min for my son to work through this problem, including a couple of false starts. But he gets to the end, which made me really happy: