Let sin- - -5-13 and cos- - -3-5 where - - - 0- -4 then tan 2-

Determine the value of tan 2α.Step 1: Calculate the value of tan(α +β ) andtan(α β ) sin(α β )=5/130ex0ex∴cos(α β )=12/13Similarly cos(α +β )=3/5∴sin(α +β )= ...

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Standard XII

Mathematics

Algebra of Derivatives

Let sinα -β =...

Question

Let $\sin (α - β) = \frac{5}{13}$ and $\cos (α + β) = \frac{3}{5}$ where $α, β \in (0, \frac{π}{4})$ then $\tan 2 α$

$\frac{63}{16}$

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$\frac{61}{16}$

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$\frac{65}{16}$

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$\frac{32}{9}$

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Solution

The correct option is A
$\frac{63}{16}$

Determine the value of $\tan 2 α$ .
Step 1: Calculate the value of $\tan (α + β) and \tan (α - β)$
$\sin (α - β) = \frac{5}{13} ∴ \cos (α - β) = \frac{12}{13}$
Similarly $\cos (α + β) = \frac{3}{5}$
$∴ \sin (α + β) = \frac{4}{5}$
Thus, $\tan (α - β) = \frac{5}{12} and \tan (α + β) = \frac{4}{3}$
Step 2:Calculate the value of $\tan 2 α$
$\Rightarrow \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \Rightarrow \tan (α + β + α - β) = \frac{\tan (α + β) + \tan (α - β)}{1 - \tan (α + β) \tan (α - β)} \Rightarrow \tan (2 α) = \frac{\frac{4}{3} + \frac{5}{12}}{1 - (\frac{4}{3} . \frac{5}{12})} \Rightarrow \tan (2 α) = \frac{63}{16}$
Hence, option A is the correct answer.

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