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  4. If cos(α+β)=(4/5), sin (α-β)=(5/13) and α, β lie between 0 and π/4 find tan 2α

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Q. If $cos(\alpha+\beta)=\frac{4}{5}, sin (\alpha-\beta)=\frac{5}{13} and \alpha, \beta lie between 0 and \pi/4 find tan 2\alpha$

IIT JEEIIT JEE 1979

Solution:

Since, $ cos(\alpha+\beta)=\frac{4}{5}$
$and sin(\alpha+\beta)=\frac{5}{13}$
$\therefore tan(\alpha+\beta)=\frac{3}{4}and tan (\alpha-\beta)=\frac{5}{12}$
Now, tan 2 \alpha = tan [(\alpha + \beta) + (\alpha - \beta)]
$=\frac{tan (\alpha + \beta)+ tan (\alpha -\beta )}{1 - tan ( \alpha + \beta) tan ( \alpha - \beta )}=\frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}.\frac{5}{12}}=\frac{56}{33}$