How to prove sin(θ+ϕ)cos(θ−ϕ)=tanθ+tanϕ1+tanθtanϕ?
3 Answers
Please see the proof below
Explanation:
We need
Therefore,
Dividing by all the terms by
See Explanation
Explanation:
Let
Dividing by
Dividing by
hence proved.
Explanation:
using the trigonometric identities
∙xsin(x+y)=sinxcosy+cosxsiny
∙xcos(x−y)=cosxcosy+sinxsiny
consider the left side
=sinθcosϕ+cosθsinϕcosθcosϕ+sinθsinϕ
divide terms on numerator/denominator by cosθcosϕ
and cancel common factors
=sinθcosϕcosθcosϕ+cosθsinϕcosθcosϕcosθcosϕcosθcosϕ+sinθsinϕcosθcosϕ=sinθcosθ+sinϕcosϕ1+sinθcosθ×sinϕcosϕ
=tanθ+tanϕ1+tanθtanϕ
=right side ⇒verified