Question

How do you prove `(sin2x + sin2y)/(cos2x + cos2y) = tan(x + y)`?

Answer 1

With the sum identities

• `sin(x) + sin(y) = 2sin((x+y)/2)cos((x-y)/2)`

• `cos(x) + cos(y) = 2cos((x+y)/2)cos((x-y)/2)`

we have

`(sin(2x) + sin(2y))/(cos(2x) + cos(2y)) = (2sin((2x+2y)/2)cos((2x - 2y)/2))/(2cos((2x+2y)/2)cos((2x - 2y)/2))`

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

`= tan(x+y)`

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In the case that the sum identities above are not given, here is a short proof of them using the angle-addition formulas:

Let `a = (x+y)/2` and `b = (x-y)/2`

Then

`sin(x) + sin(y) = sin(a + b) + sin(a-b)`

`= sin(a)cos(b) + cos(a)sin(b) + sin(a)cos(b) - cos(a)sin(b)`

`= 2sin(a)cos(b)`

`= 2sin((x+y)/2)cos((x-y)/2)`

And

`cos(x) + cos(y) = cos(a+b) + cos(a-b)`

`= cos(a)cos(b) - sin(a)sin(b) + cos(a)cos(b) + sin(a)sin(b)`

`= 2cos(a)cos(b)`

`= 2cos((x+y)/2)sin((x-y)/2)`