Question

# If $\tan 2A=\cot (A-{{18}^{\circ }})$, where $2A$ is an acute angle, then find the value of $A$.

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Hint: Since, $2A$ is an acute angle so we can say that $A$ will also be an acute angle. Thus, all the trigonometric equations applied at this angle will be normal and basic. We will have to convert $\tan 2A$ in the form of the $\cot 2A$ directly, to easily equate both sides of the equation.

Here, we have the following given equation as
$\Rightarrow \tan 2A=\cot (A-{{18}^{\circ }})...\text{ (1)}$
We have to convert the $\tan 2A$ in the $\cot 2A$ form.
As per question, $2A$ is an acute angle, then we can say that $A$ will also be an acute angle,
And from trigonometric complementary equations, we have
$\Rightarrow \tan \theta =\cot \left( {{90}^{\circ }}-\theta \right)$
Substituting this value of $\tan \theta$ in terms of $\cot \theta$ in equation (1), we get
\begin{align} & \Rightarrow \tan 2A=\cot \left( {{90}^{\circ }}-2A \right) \\ & \Rightarrow \tan 2A=\cot \left( A-{{18}^{\circ }} \right) \\ \end{align}
From both of the above equations, we get
$\Rightarrow \cot \left( {{90}^{\circ }}-2A \right)=\cot \left( A-{{18}^{\circ }} \right)...\text{ (2)}$
Thus, comparing the angles from both of the sides of the equation (2), we get
$\Rightarrow {{90}^{\circ }}-2A=A-{{18}^{\circ }}$
Transposing terms from either side of the equation to keep similar variables at same side, we get
\begin{align} & \Rightarrow {{90}^{\circ }}-2A=A-{{18}^{\circ }} \\ & \Rightarrow -2A-A=-{{90}^{\circ }}-{{18}^{\circ }} \\ & \Rightarrow -3A=-{{108}^{\circ }} \\ \end{align}
Now, cancelling out the negation from both sides, we get
\begin{align} & \Rightarrow -3A=-{{108}^{\circ }} \\ & \Rightarrow 3A={{108}^{\circ }} \\ \end{align}
With cross-multiplication in the above equation, we get
\begin{align} & \Rightarrow 3A={{108}^{\circ }} \\ & \Rightarrow A=\dfrac{{{108}^{\circ }}}{3} \\ & \Rightarrow A={{36}^{\circ }} \\ \end{align}
Hence, The value of acute angle $A={{36}^{\circ }}$.

Note: A probable mistake which can be made here is that, while conversion of one of the trigonometric functions to another form, there could be a sign conventional mistake irrespective of the given information about the acute angle.