Friday, September 30, 2016

trigonometry - Trigonometric identity involving double angles



If α and β are acute angles and cos2α=3cosβ13cos2β, then prove that tanα=2tanβ.



I tried this question by taking the formula of cos2α in terms of tan (which is of degree two) but I couldn't prove it. Please suggest some hints.


Answer



Using Weierstrass substitution in either sides.




1tan2α1+tan2α=31tan2β1+tan2β131tan2β1+tan2β



1tan2α1+tan2α=24tan2β2+4tan2β



Using Componendo and dividendo, tan2α1=4tan2β2


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