Question

How do you solve $2\tan x = \sin 2x$ ?

Answer 1

To solve $2\tan x = \sin 2x$, we will first write $\tan x$ in terms of $\sin x$ and $\cos x$. As we know $\tan x = \dfrac{{\sin x}}{{\cos x}}$ and from double angle trigonometric formula we can write $\sin 2x = 2\sin x\cos x$, using this we will rewrite the given equation. Then we will simplify it and use the general solution condition to find the result.

Complete step by step answer:
Given equation is $2\tan x = \sin 2x - - - (1)$.
As we know from the double angle formula: $\sin 2x = 2\sin x\cos x$.
Also, we know that $\tan x = \dfrac{{\sin x}}{{\cos x}}$.
On putting this value in $(1)$, we get
$\Rightarrow 2\dfrac{{\sin x}}{{\cos x}} = 2\sin x\cos x$
Taking all the terms to the left-hand side of the equation, we get
$\Rightarrow 2\dfrac{{\sin x}}{{\cos x}} - 2\sin x\cos x = 0$
Taking $2$ common, we get
$\Rightarrow 2\left( {\dfrac{{\sin x}}{{\cos x}} - \sin x\cos x} \right) = 0$
Dividing both the sides by $2$, we get
$\Rightarrow \dfrac{{\sin x}}{{\cos x}} - \sin x\cos x = 0$

Taking $\sin x$ common, we get
$\Rightarrow \sin x\left( {\dfrac{1}{{\cos x}} - \cos x} \right) = 0$
On simplifying, we get
$\Rightarrow \dfrac{{\sin x\left( {1 - {{\cos }^2}x} \right)}}{{\cos x}} = 0$
On solving, we get
$\Rightarrow \sin x = 0$ or $1 - {\cos ^2}x = 0$
$\Rightarrow \sin x = 0$ or ${\cos ^2}x = 1$
On further simplification, we get
$\Rightarrow \sin x = 0$ or $\cos x = \pm 1$
General solution of $\sin x = 0$ is $x = n\pi$, where $n \in Z$. Also, general solution of $\cos x = 1$ is $x = 2n\pi$, where $n \in Z$and general solution of $\cos x = - 1$ is $x = \left( {2n + 1} \right)\pi$, where $n \in Z$. On combining, the general solution of $\cos x = \pm 1$ is $x = n\pi$, where $n \in Z$.

Therefore, the solution of $2\tan x = \sin 2x$ is $x = n\pi$ where $n \in Z$.

Note: An equation involving two or more trigonometric ratios of an unknown angle is called a trigonometric equation. A trigonometric equation is different from a trigonometric identity. An identity is satisfied for every value of the unknown angle whereas a trigonometric equation is satisfied for some particular values of the unknown angles.