If-sin-A-B-1-10-cos-A-B-2-29-0-lt-A-lt-pi-4-0-lt-B-lt-pi-4-Find-tan-2A-

Question Number 164391 by bobhans last updated on 16/Jan/22

If {\begin{cases} \sin (A - B) = \frac{1}{\sqrt{10}} \\ \cos (A + B) = \frac{2}{\sqrt{29}} \end{cases}; 0 < A < \frac{π}{4}; 0 < B < \frac{π}{4}

Find \tan 2 A .

Answered by cortano1 last updated on 16/Jan/22

\begin{matrix} \sin (A - B) = \frac{1}{\sqrt{10}} \\ \cos (A + B) = \frac{2}{\sqrt{29}} \end{matrix}} \Rightarrow {\begin{cases} 0 < A + B < \frac{π}{2} \\ - \frac{π}{4} < A - B < \frac{π}{4} \end{cases}

{\begin{cases} \tan (A - B) = \frac{1}{3} \\ \tan (A + B) = \frac{5}{2} \end{cases}

\Rightarrow \tan 2 A = \frac{\tan (A + B) + \tan (A - B)}{1 - \tan (A + B) . \tan (A - B)}

\tan 2 A = \frac{\frac{5}{2} + \frac{1}{3}}{1 - \frac{5}{6}} = 17

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