First, we'll write the tangent
identity:
tan(a+b) =
sin(a+b)/cos(a+b)
We'll write the formulas for the sine and
cosine of the sum of angles a and b:
sin(a+b) = sina*cosb +
sinb*cosa
cos(a+b) = cosa*cosb -
sina*sinb
We'll substitute sin(a+b) and cos(a+b) by their
formulas:
tan(a+b) = (sina*cosb + sinb*cosa)/(cosa*cosb -
sina*sinb)
We'll factorize by
cosa*cosb:
tan(a+b) =cosa*cosb*[(sina*cosb/cosa*cosb) +
(sinb*cosa/cosa*cosb)]/cosa*cosb*[1 -
(sina*sinb/cosa*cosb)]
We'll simplify and we'll
get:
tan(a+b) = (sina/cos a + sinb/cos b)/(1 - tan a*tan
b)
tan(a+b) = (tan a + tan b)/(1 - tan a*tan
b) q.e.d.
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