Let's write this out fully:

a^2 / (a*sin(c))^2 - b^2 / (b*sin(c))^2 = 1

a^2 / (a^2*sin^2(c)) - b^2 / (b^2*sin^2(c)) = 1

The a^2 and b^2's cancel out

1/sin^2(c) - 1/tan^2(c) = 1

1/sin^2(c) - cos^2(c)/sin^2(c) = 1

Under the same denominator...

(1 - cos^2(c)) / sin^2(c) = 1

Since we know sin^2(c) + cos^2(c) = 1 (Pythagorean identity),

1 - cos^2(c) = sin^2(c)

Thus, we have :

sin^2(c) / sin^2(c) = 1

1=1