I was attempting to solve a geometrical problem the other day, a problem which, due to my complete misunderstanding, had no solution, when this popped out. It is probably bog-standard, but new to me, and this time I don’t have the heart to check if it is one of the 100 proofs of The Pythagoras theorem.

Now let theta be the angle ACB. Angle ABD is then 2*theta.

Set r = 1, then a is sin(2*theta) and b is cos(2*theta), and so

sin(theta) = (1 – b)/a = 1/a -b/a = cosec(2*theta) – cot(2*theta)

and cosec(theta) = (1 + b)/a = 1/a + b/a = cosec(2*theta) + cot(2*theta)

I’ve done most of the work,

so now **you can show** that sin(2*theta) = 2*sin(theta)*cos(theta)