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  • If cos (α+β)=4/5 and \sin (α+β)=5/13 where α lie between 0 and π/4, find value of tan 2α [Hint: Express tan 2α as tan (α + β + α – β]

If \cos (\alpha+\beta)=\frac{4}{5} \text { and } \sin (\alpha-\beta)=\frac{5}{13} where α lie between 0 and π/4, find value of tan 2α
[Hint: Express tan 2 \: \: \alpha \: \: as\: \: tan ( \alpha + \beta + \alpha - \beta ]

Answers (1)

\\ \cos \left( \alpha + \beta \right) =\frac{4}{5}\text{~ so,}\tan \left( \alpha + \beta \right) =\frac{3}{4}~~~~~ \\\\ \text{~~~~ And}\sin \left( \alpha - \beta \right) =\frac{5}{13}~~\tan \left( \alpha - \beta \right) =\frac{5}{12}~ \\\\ tan2 \alpha =\tan \left[ \alpha + \beta + \alpha - \beta \right] ~~ \\\\
    \\ =\tan \left[ \left( \alpha + \beta \right) + \left( \alpha - \beta \right) \right] ~ \\\\ =\frac{\tan \left( \alpha + \beta \right) +\tan \left( \alpha - \beta \right) }{1-\tan \left( \alpha + \beta \right) \tan \left( \alpha - \beta \right) } \\\\

 

   \\ =~ \frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}\ast\frac{5}{12}}=\frac{56}{33}~ \\\\

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