Verify sin(x-y)/sin(x+y)=tan(x)-tan(y)/tan(x)+tan(y)

Answer 1

The sine angle addition/subtraction identities are:

sin(x – y) = sin(x)cos(y) – cos(x)sin(y)

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

So sin(x-y)/sin(x+y) = [sin(x)cos(y) – cos(x)sin(y)]/[sin(x)cos(y) + cos(x)sin(y)]

Divide both the numerator and denominator by cos(x):

[sin(x)cos(y) – cos(x)sin(y)]/cos(x) / [sin(x)cos(y) + cos(x)sin(y)]/cos(x) Note: This is essentially multiplying by sec(x)/sec(x)

Now, split up the numerator and the denominator:

[sin(x)cos(y)/cos(x) - cos(x)sin(y)/cos(x)] / sin(x)cos(y)/cos(x) + cos(x)sin(y)/cos(x)]

Now simplify sin(x)/cos(x) as tan(x):

[tan(x)cos(y) - sin(y)] / tan(x)cos(y) + sin(y)]

Now divide both numerator and denominator by cos(y):

[tan(x)cos(y) - sin(y)]/cos(y) / tan(x)cos(y) + sin(y)]/cos(y)

Again, split the pieces of the numerator and denominator apart:

[tan(x)cos(y)/cos(y) - sin(y)/cos(y)] / [tan(x)cos(y)/cos(y) + sin(y)/cos(y)]

Simplify:

[tan(x) - tan(y)]/[tan(x) + tan(y)]