how would you prove that sin^2(a)-cos^2(b)= sin^2(b)-cos^2(a).

Question

i'm not completely sure that this is right but i used the difference of two squares on it to get (sin(a)+cos(b))(sin(a)-cos(b)) then after that i am stuck. please help

Answer 1

It will surprise you to see how easy this one is.

we know that sin^2 a + cos^2 a = 1
and sin^2 b + cos^2 b = 1
then
sin^2 a + cos^2 a = sin^2 b + cos^2 b
now let's "move" the terms cos^2 a and cos^2 b to get the needed equation

sin^2 a - cos^2 b = sin^2 b - cos^2 a

Done!