Evaluate sin(2tan⁻¹(2/3)) + cos(tan⁻¹√3)

Evaluate \( \sin\left(2\tan^{-1}\left(\frac{2}{3}\right)\right) + \cos\left(\tan^{-1}\sqrt{3}\right) \)

Solution:

Let

\[ \theta = \tan^{-1}\left(\frac{2}{3}\right) \]

Then,

\[ \tan \theta = \frac{2}{3} \]

Using identity:

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

\[ = \frac{2 \cdot \frac{2}{3}}{1 + \left(\frac{2}{3}\right)^2} \]

\[ = \frac{4/3}{1 + 4/9} \]

\[ = \frac{4/3}{13/9} \]

\[ = \frac{4}{3} \times \frac{9}{13} \]

\[ = \frac{12}{13} \]

Now, let

\[ \phi = \tan^{-1}(\sqrt{3}) \]

Then,

\[ \tan \phi = \sqrt{3} \Rightarrow \phi = \frac{\pi}{3} \]

So,

\[ \cos(\phi) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Therefore,

\[ \sin(2\theta) + \cos(\phi) = \frac{12}{13} + \frac{1}{2} \]

\[ = \frac{24 + 13}{26} \]

\[ = \frac{37}{26} \]

\[ \sin\left(2\tan^{-1}\left(\frac{2}{3}\right)\right) + \cos\left(\tan^{-1}\sqrt{3}\right) = \frac{37}{26} \]

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