Grade 11Trigonometry
If tan β = n sin α cos α/1-n sin^2 α , prove that tan (α - β) = (1- n) tan α
If tan β = n sin α cos α/1-n sin^2 α , prove that tan (α - β) = (1- n) tan α
1 Answer
Aditya Gupta
thanks 4 askin ananya.
tan (α - β) = [tanα - tanβ]/[1+tanαtanβ]....(1)
but tan β = n sin α cos α/1-n sin^2α.......(2) ,
we see that sin α cos α=sin2α/2=tanα/[1+tan^2α] and sin^2α= tan^2α/[1+tan^2α] . now substitute these values in (2) and simplify, we get
tanβ= ntanα/[1+tan^2α – ntan^2α]
substitute this value in (1), and simplify, we have
tan (α - β) = tan (α)[1+tan^2α – ntan^2α – n]/[1+tan^2α]
factoring [1+tan^2α – ntan^2α – n] as (1-n)(1+tan^2α) we have
=tanα (1-n)(1+tan^2α)/(1+tan^2α)
= (1- n) tan α
hence proved
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