Given tan ⁡ β = n sin ⁡ α cos ⁡ α 1 − n...

Jaydan Aguirre

Jaydan Aguirre

Answered

2022-07-12

Given tan β = n sin α cos α 1 n sin 2 α , show that tan ( α β ) = ( 1 n ) tan α

Answer & Explanation

kawiarkahh

kawiarkahh

Expert

2022-07-13Added 15 answers

As we need to eliminate β ,, write tan β = tan { α ( α β ) } and expand.
For the RHS,
n sin α cos α 1 n sin 2 α = n sin α cos α cos 2 α 1 n sin 2 α cos 2 α = n tan 2 α 1 + ( 1 n ) tan 2 α
Kaeden Hoffman

Kaeden Hoffman

Expert

2022-07-14Added 3 answers

With
tan β = n sin α cos α 1 n sin 2 α
we write
n = tan β sin α cos α + tan β sin 2 α
so
1 n = sin α cos α + tan β sin 2 α tan β sin α cos α + tan β sin 2 α = sin α cos α cos 2 α tan β sin α cos α + tan β sin 2 α = tan α tan β tan α + tan β tan 2 α = 1 tan α tan ( α β )

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