The Value of \[cot[{{\operatorname{cosec}}^{-1}}\left( \dfrac{5}{3} \right)+{{\tan }^{-1}}\left( \dfrac{2}{3} \right)]is\]
A. $\dfrac{3}{17}$
B. $\dfrac{4}{17}$
C. \[\dfrac{5}{17}\]
D. $\dfrac{6}{17}$

Hint: As we know that trigonometry is one of the most significant branches in mathematics that finds huge use in distinct fields which predominantly deals with the study of the association between the sides and angles of the right-angle triangle. Hence, it helps to determine the misplaced or unbeknown sides or angles of a right triangle by applying trigonometric formulas.
To solve the given equation, we need to use trigonometric identities.

Formula Used:
In order to solve this problem we will use the formulas of trigonometry like, ${\operatorname{cosec} ^2}A{\text{ }}-{\text{ co}}{{\text{t}}^2}A{\text{ }} = {\text{ }}1$then we will put these values in the given equation and solve.
Then we have to use the formulas, $\cot B{\text{ }} = {\text{ }}\dfrac{1}{{\tan B}}{\text{ }}$and $\cot (A + B) = \dfrac{{\cot A\cot B - 1}}{{\cot B + \cot A}}$.

Complete step by step Solution:
$\cot \left[ {{{\operatorname{cosec} }^{ - 1}}\left( {\dfrac{5}{3}} \right) + {{\tan }^{ - 1}}\left( {\dfrac{2}{3}} \right)} \right]\,\,\,....(i)$
First, consider the equation, ${\operatorname{cosec} ^{ - 1}}\left( {\dfrac{5}{3}} \right){\text{ }} = {\text{ }}A$
$\operatorname{cosec} A = \left( {\dfrac{5}{3}} \right){\text{ }}$
Now, using the trigonometry identity, ${\operatorname{cosec} ^2}A{\text{ }}-{\text{ co}}{{\text{t}}^2}A{\text{ }} = {\text{ }}1$,
Putting the value of $\operatorname{cosec} $ in the above identity, then we have:
 $\cot A = \dfrac{4}{3}$
Again, take $ta{n^{ - 1}}\left( {\dfrac{2}{3}} \right)$ for $B$in the equation $(i)$, we have:
 $\tan B = \dfrac{2}{3}$
As we know that ${\text{cot }}B = \dfrac{1}{{\tan B}}{\text{ }}$, then:
${\text{cot }}B = \dfrac{3}{2}$
Since $\cot (A + B) = \dfrac{{\cot A\cot B - 1}}{{\cot B + \cot A}}$ taking it as equation$...(ii)$
Now, putting the value of $\cot A$ and $\cot B$ in equation $(ii)$
$\left[ {\dfrac{{\left( {\dfrac{4}{3}} \right)\left( {\dfrac{3}{2}} \right) - 1}}{{\left( {\dfrac{4}{3}} \right) + \left( {\dfrac{3}{2}} \right)}}} \right] = \cot (A + B)$
After simplifying the above equation, we obtain:
$\dfrac{{(2 - 1) \times 6}}{{8 + 9}} = \dfrac{6}{{17}}$
Therefore, the value of $\cot \left[ {{{\operatorname{cosec} }^{ - 1}}\left( {\dfrac{5}{3}} \right) + {{\tan }^{ - 1}}\left( {\dfrac{2}{3}} \right)} \right]$is $\dfrac{6}{{17}}$.

Hence, the correct option is (D).

Note: While solving the trigonometric equation, we must know when to use the right trigonometric identity and also know about trigonometric angles, it is easy to memorize and recall, also if basic trigonometric identities knowledge is also needed for understanding the question, we also know that the trigonometric angles.