Question

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

Answer 1

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.

“Complete step-by-step answer:”
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,

& \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{aligned}$$ Let us take $$\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{aligned} & =2\cos \left( \dfrac{6x}{2} \right)\cos \left( \dfrac{2x}{2} \right) \\\ & =2\cos 3x\cos x \\\ \end{aligned}$$ Now let us substitute the values of $$\left( \cos 4x+\cos 2x \right)$$ and $$\left( \sin 4x+\sin 2x \right)$$ in the LHS. $$\begin{aligned} & 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{aligned}$$ Take $$\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.