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# Prove the following : $\dfrac{\cos 4x+\cos 3x+\cos 2x}{\sin 4x+\sin 3x+\sin 2x}=\cot 3x$

Last updated date: 29th Mar 2023
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Hint: Take the LHS of the expression. Apply the basic trigonometric identities in the numerator and denominator of the expression and simplify it. Hence, prove that LHS=RHS.

We have been given the expression, $\dfrac{\cos 4x+\cos 3x+\cos 2x}{\sin 4x+\sin 3x+\sin 2x}=\cot 3x$.
Let us consider the LHS of the expression.
LHS = $\dfrac{\cos 4x+\cos 3x+\cos 2x}{\sin 4x+\sin 3x+\sin 2x}$.
We know the basic trigonometric identities,
\begin{align} & \cos a+\cos b=2\cos \left( \dfrac{a+b}{2} \right)\cos \left( \dfrac{a-b}{2} \right) \\ & \sin a+\sin b=2\sin \left( \dfrac{a+b}{2} \right)\cos \left( \dfrac{a-b}{2} \right) \\ \end{align}
Let us take $\left( \cos 4x+\cos 2x \right)$ from the numerator and apply the trigonometric identity.
$\therefore \cos 4x+\cos 2x=2\cos \left( \dfrac{4x+2x}{2} \right)\cos \left( \dfrac{4x-2x}{2} \right)$
\begin{align} & =2\cos \left( \dfrac{6x}{2} \right)\cos \left( \dfrac{2x}{2} \right) \\ & =2\cos 3x\cos x \\ \end{align}
Now let us substitute the values of $\left( \cos 4x+\cos 2x \right)$ and $\left( \sin 4x+\sin 2x \right)$ in the LHS.
\begin{align} & LHS=\dfrac{\left( \cos 4x+\cos 2x \right)+\cos 3x}{\left( \sin 4x+\sin 2x \right)+\sin 3x} \\ & LHS=\dfrac{2\cos 3x\cos x+\cos 3x}{2\sin 3x\cos x+\sin 3x} \\ \end{align}
Take $\left( \cos 3x \right)$common from the numerator and $\left( \sin 3x \right)$from the denominator.
$=\dfrac{\cos 3x\left[ 2\cos x+1 \right]}{\sin 3x\left[ 2\cos x+1 \right]}$
Cancel out $\left( 2\cos x+1 \right)$ from the numerator and denominator.
$\therefore LHS=\dfrac{\cos 3x}{\sin 3x}$
We know that, $\dfrac{\cos x}{\sin x}=\cot x$
Hence, $\dfrac{\cos 3x}{\sin 3x}=\cot 3x$.
$\dfrac{\cos 4x+\cos 3x+\cos 2x}{\sin 4x+\sin 3x+\sin 2x}=\cot 3x$
$\therefore$LHS = RHS
Hence proved.

Note: Remember the basic trigonometric identities like $\left( \sin a+\sin b \right)$ and $\left( \cos a+\cos b \right)$ which we have used to solve this expression. Trigonometric identities are an important section in maths. Just apply the formula and simplify it to get the required answer.