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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions1 Mark
MCQ
$\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right)=$
  • A
    $\frac{17}{7}$
  • B
    $-\frac{17}{7}$
  • C
    $\frac{7}{17}$
  • ✓
    $-\frac{7}{17}$
✓

Answer

Correct option: D.
$-\frac{7}{17}$
(d) $-\frac{7}{17}$$\left[\right.$ Hint : $2 \tan ^{-1}\left(\frac{1}{5}\right)=\tan ^{-1}\left[\frac{2 \times \frac{1}{5}}{1-\left(\frac{1}{5}\right)^2}\right]=\tan ^{-1}\left(\frac{5}{12}\right)$
$\therefore 2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}=\tan ^{-1}\left(\frac{5}{12}\right)-\tan ^{-1}(1)$
$=\tan ^{-1}\left[\frac{\frac{5}{12}-1}{1+\frac{5}{12} \times 1}\right]=\tan ^{-1}\left(\frac{-7}{17}\right)$
$\begin{array}{r}\therefore \tan \left[2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right]=\tan \left[\tan ^{-1}\left(-\frac{7}{17}\right)\right] \\ \left.=-\frac{7}{17} .\right]\end{array}$

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