Question

How do you prove $\sin 2x=\left( \tan x \right)\left( 1+\cos 2x \right)$?

Answer 1

We solve this question by first considering the RHS of the given equation and prove it equal to the LHS. We will simplify the RHS of the equation and get the value equal to the LHS. In order to simplify RHS we will use the trigonometric identities and formulas which are given as:
$\begin{aligned} & 1+\cos 2x=2{{\cos }^{2}}x \\\ & \tan x=\dfrac{\sin x}{\cos x} \\\ \end{aligned}$
$2\sin x\cos x=\sin 2x$

Complete step-by-step solution:
We have been given an equation $\sin 2x=\left( \tan x \right)\left( 1+\cos 2x \right)$.
We have to prove that LHS=RHS.
In order to prove let us first consider the RHS of the given equation. Then we will get
$\Rightarrow \left( \tan x \right)\left( 1+\cos 2x \right)$
Now, we know that $1+\cos 2x=2{{\cos }^{2}}x$
Substituting the value in the above equation we will get
$\Rightarrow \tan x\times 2{{\cos }^{2}}x$
Now, we know that $\tan x=\dfrac{\sin x}{\cos x}$
Substituting the value in the above equation we will get
$\Rightarrow \dfrac{\sin x}{\cos x}\times 2{{\cos }^{2}}x$
Now, simplifying the above equation we will get
$\Rightarrow 2\sin x\cos x$
Now, we know that $2\sin x\cos x=\sin 2x$
Substituting the value in the above equation we will get
$\Rightarrow \sin 2x$ which is equal to the LHS.
Hence we get LHS=RHS
Hence proved

Note: To solve this type of question students must have knowledge of trigonometric identities and formulas. As there are many formulas in the trigonometry students can use other formulas also. Students must be careful while using the formulas because they may be confused. We can also solve the question by taking LHS but in this particular question taking LHS and solving further is quite lengthy.