The value of <span class="math-tex">\(\cot \left(\text{cosec}^{-1} \dfrac{5}{3}+ \tan^{-1} \dfrac{2}{3}\right)\)</span> is
Let \(\theta_1 = \text{cosec}^{-1}\frac{5}{3}\). Then \(\sin\theta_1 = \frac{3}{5}\), so \(\tan\theta_1 = \frac{3}{4}\).
\(\tan\left(\theta_1 + \tan^{-1}\frac{2}{3}\right) = \frac{\frac{3}{4} + \frac{2}{3}}{1 - \frac{3}{4} \cdot \frac{2}{3}} = \frac{17/12}{1/2} = \frac{17}{6}\)
The cot is the reciprocal: \(\frac{6}{17}\).
Answer: (A) \(\frac{6}{17}\)
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