Proof:
(Note: ctg(a) = cos(a)/sin(a))
Firstly, let's simplify the LHS of the equation: (cos(a+b) + cos(a-b)) / (sin(a+b) + sin(a-b))
Expand the cosine terms= (cos(a)cos(b) - sin(a)sin(b) + cos(a)cos(b) + sin(a)sin(b)) / (sin(a)cos(b) + cos(a)sin(b) + sin(a)cos(b) - cos(a)sin(b))
= 2cos(a)cos(b) / 2sin(a)cos(b) = ctg(a)
Therefore, (cos(a+b) + cos(a-b)) / (sin(a+b) + sin(a-b)) = ctg(a)
Proof:
(Note: ctg(a) = cos(a)/sin(a))
Firstly, let's simplify the LHS of the equation: (cos(a+b) + cos(a-b)) / (sin(a+b) + sin(a-b))
Expand the cosine terms
= (cos(a)cos(b) - sin(a)sin(b) + cos(a)cos(b) + sin(a)sin(b)) / (sin(a)cos(b) + cos(a)sin(b) + sin(a)cos(b) - cos(a)sin(b))
= 2cos(a)cos(b) / 2sin(a)cos(b) = ctg(a)
Therefore, (cos(a+b) + cos(a-b)) / (sin(a+b) + sin(a-b)) = ctg(a)