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If $3\cos \theta -4\sin \theta =2\cos \theta +\sin \theta $, is given, then find the value of $\tan \theta $.

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Hint:In order to find the solution of this question, we will start from the given equation and then we will try to form $\tan \theta $, by using the formula, $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$ and then we will calculate the value of $\tan \theta $.

Complete step-by-step answer:
In this question, we have been asked to find the value of $\tan \theta $, when it is given that $3\cos \theta -4\sin \theta =2\cos \theta +\sin \theta $. To solve this, we will first consider the given equality, that is, $3\cos \theta -4\sin \theta =2\cos \theta +\sin \theta $. We will try to form $\dfrac{\sin \theta }{\cos \theta }$ here. For that, we will write the terms of $\cos \theta $ on the left hand side and the terms of $\sin \theta $ on the right hand side. So, we can write the given equation $3\cos \theta -4\sin \theta =2\cos \theta +\sin \theta $ as,
$3\cos \theta -2\cos \theta =\sin \theta +4\sin \theta $
We know that arithmetic operations are applied to like terms. So, we get the above equation as,
$\cos \theta =5\sin \theta $
Now, we will divide the whole equation by $\cos \theta $. On doing so, we get,
$\dfrac{\cos \theta }{\cos \theta }=\dfrac{5\sin \theta }{\cos \theta }$
We know that the common terms of the numerator and the denominator gets cancelled, so we get,
$1=\dfrac{5\sin \theta }{\cos \theta }$
We will now divide the equation by 5. So, we get,
$\begin{align}
  & \dfrac{1}{5}=\dfrac{5\sin \theta }{5\cos \theta } \\
 & \Rightarrow \dfrac{\sin \theta }{\cos \theta }=\dfrac{1}{5} \\
\end{align}$
We also know that $\dfrac{\sin \theta }{\cos \theta }$ can be expressed as $\tan \theta $. So, applying that, we get,
$\tan \theta =\dfrac{1}{5}$
Hence, we can say that $\tan \theta =\dfrac{1}{5}$, if $3\cos \theta -4\sin \theta =2\cos \theta +\sin \theta $.

Note: While solving the question, the possible mistakes that can be made are in the calculations. Also, we have to remember that $\dfrac{\sin \theta }{\cos \theta }$ can be expressed as $\tan \theta $. So, we should try to form $\tan \theta $, either by taking the terms to one side and then dividing them by $\cos \theta $.
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