I came across this interesting Trig problem that involves an angle θ formed by two rays OB and OA. there are two circles, one with radius a and one with radius b, on on the ray OA that are both tangent to one another. The ray OB is tangent to both of the circles.
Show that cos(θ)=(ab)^(1/2)/(a+b)/2, in other words that cosine is equal to the ratio of the geometric mean to the arithmetic mean.
The Attempt at a Solution
I understand that going from sin to cos is just a matter of applying the pythagorean identity, what I'm having trouble understanding is why sin(θ)= (b-a)/(b+a).
Thanks for the help.