Question

If sin-1(x/a) + sin-1(y/b) = sin-1(c^2/ab) then show that b^2x^2 + 2xy(a^2b^2 + c4) + a^2y^2 = c^4.

Answer 1

use the sum of sines formula:

sin(A+B) = sinA cosB + cosA sinB
We have
A = arcsin(x/a), so cosA = √(a^2-x^2)/a
B = arcsin(y/b), so cosB = √(b^2-y^2)/b
C = arcsin(c^2/ab)

That gives us sin(A+B) = sinC

(x/a)(√(b^2-y^2)/b) + (√(a^2-x^2)/a)(y/b) = c^2/ab

Hmmm. That's getting complicated. There is clearly some tricky insight needed. I'll think on it. Need to relate a,b,c in some way.

Answer 2

Thanks for your help, its solved.