Evaluate tan{2tan⁻¹(1/5) − π/4}

Evaluate \( \tan\left(2\tan^{-1}\left(\frac{1}{5}\right) – \frac{\pi}{4}\right) \)

Solution:

Let

\[ \theta = \tan^{-1}\left(\frac{1}{5}\right) \]

Then,

\[ \tan \theta = \frac{1}{5} \]

Using identity:

\[ \tan(2\theta) = \frac{2\tan\theta}{1 – \tan^2\theta} \]

\[ = \frac{2 \cdot \frac{1}{5}}{1 – \left(\frac{1}{5}\right)^2} \]

\[ = \frac{2/5}{1 – 1/25} \]

\[ = \frac{2/5}{24/25} \]

\[ = \frac{2}{5} \times \frac{25}{24} \]

\[ = \frac{5}{12} \]

Now evaluate:

\[ \tan\left(2\theta – \frac{\pi}{4}\right) \]

Using identity:

\[ \tan(A – B) = \frac{\tan A – \tan B}{1 + \tan A \tan B} \]

\[ = \frac{\frac{5}{12} – 1}{1 + \frac{5}{12} \cdot 1} \]

\[ = \frac{\frac{5}{12} – \frac{12}{12}}{1 + \frac{5}{12}} \]

\[ = \frac{-7/12}{17/12} \]

\[ = -\frac{7}{17} \]

\[ \tan\left(2\tan^{-1}\left(\frac{1}{5}\right) – \frac{\pi}{4}\right) = -\frac{7}{17} \]

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