Given:
sin(a+b)/(cos(a)*cos(b)) = tan(a) + tan(b)
Let's manipulate the more complicated left side until it looks like the simpler right side!
Rewrite numerator using sum and difference formula:
(sin(a)cos(b) + sin(b)cos(a))/(cos(a)*cos(b)) = tan(a) + tan(b)
Split up the fraction and simplify each chunk:
sin(a)cos(b)/(cos(a)*cos(b)) + sin(b)cos(a)/(cos(a)*cos(b)) = tan(a) + tan(b)
sin(a)/(cos(a)*cos(b)) + sin(b)/(cos(a)*cos(b)) = tan(a) + tan(b)
By the quotient identity:
sin(a)/cos(a) = tan(a)
sin(b)/cos(b) = tan(b)
Therefore,
sin(a)/(cos(a)) + sin(b)/(cos(b)) = tan(a) + tan(b).
