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Find the value of A if $\tan 2A = \cot \left( {A - {{18}^ \circ }} \right)$, where 2A is an acute angle.

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Hint: In order to solve this question we have to convert tan in terms of $\cot {\text{ }} as \left[ {{\text{ tan}}\theta = \left( {\cot {{90}^ \circ } - \theta } \right)} \right]$ by doing so we will get both sides in terms of cot.

Complete step-by-step answer:
Now, we have given that
If $\tan 2A = \cot \left( {A - {{18}^ \circ }} \right)$
And 2A is an acute angle.
Now we have to find the value of A.
Now this question is related to trigonometric.
Ratios of complementary angles,
Complementary Angles- Two angles are said to be complementary if their sum is equal to ${90^ \circ }$.
Also we know that,
$\cot \left( {{{90}^ \circ } - x} \right) = \tan x$ ------(1)
According to the given question,
$\tan 2A = \cot \left( {A - {{18}^ \circ }} \right)$
Since, $2A$ is an acute angle thus from equation(1) we get,
$\cot \left( {{{90}^ \circ } - 2A} \right) = \cot \left( {A - {{18}^ \circ }} \right)$
Now, eliminate cot from both sides, we get
${90^ \circ } - 2A = A - {18^ \circ }$
$\Rightarrow$ $3A = {108^ \circ }$
$\Rightarrow$ $A = \dfrac{{{{108}^ \circ }}}{3}$
$\Rightarrow$ $A = {36^ \circ }$
Thus, the value of A is ${36^ \circ }$.

Note: Whenever we face such types of questions, the key concept is that we must covert tan in terms of cot or vice versa. It is clearly visible that here $2A$ represents an acute angle. First we will use the identity $\tan \theta = \cot \left( {{{90}^ \circ } - \theta } \right)$ then eliminate cot (or tan) then by simplifying the equations we will get our required answer.