We have sin (a+b) = (sin a * cos b + cos a * sin b) and cos(a + b) = (cos a * cos b – sin a * sin b)
Also tan (a + b) = [sin (a+b)] / [cos (a+b)]
=> [sin a * cos b + cos a * sin b] / [cos a * cos b – sin a * sin b]
divide all the terms by cos a * cos b
=> [(sin a * cos b)/( cos a * cos b )+ (cos a * sin b)/ ( cos a * cos b )] / [(cos a * cos b)/ ( cos a * cos b ) – (sin a * sin b)/ ( cos a * cos b )]
cancel the common terms in the numerator and denominator and use sin x / cos x = tan x
=> (tan a + tan b)/(1 – tan a * tan b)
Therefore we get:
tan (a +b) = (tan a + tan b)/(1 – tan a * tan b)
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