Question

What is the value of the trigonometric expression $2\cos x - \cos 3x - \cos 5x$?
A. $16\cos^{3}x \sin^{2}x$
B. $16\sin^{3}x \cos^{2}x$
C. $4\cos^{3}x \sin^{2}x$
D. $4\sin^{3}x \cos^{2}x$

Answer 1

Hint: Simplify the given trigonometric equation using the formula of $\cos A + \cos B$. Further simplify the equation using the formulas of $\cos 2A$ and $\sin 2A$ to reach the required answer.

Formula Used:
$\cos A + \cos B = 2\cos\left( {\dfrac{{A + B}}{2}} \right) \cos\left( {\dfrac{{A - B}}{2}} \right) $
$\cos 2A = \cos^{2}A - sin^{2}A$
$\cos^{2}A + \sin^{2}A = 1$
$\sin 2A = 2\sin A \cos A$

Complete step by step solution:
 The given trigonometric equation is $2\cos x – \cos 3x – \cos 5x$.
Let $T$ be the value of the given trigonometric expression.
$T = 2\cos x -\ cos 3x - \cos 5x$
$ \Rightarrow $$T = 2\cos x - \left( {\cos 5x + \cos 3x} \right)$
Now apply the formula of $\cos A + \cos B = 2\cos\left( {\dfrac{{A + B}}{2}} \right) \cos\left( {\dfrac{{A - B}}{2}} \right) $.
$T = 2\cos x – 2\cos\left( {\dfrac{{5x + 3x}}{2}} \right) \cos\left( {\dfrac{{5x - 3x}}{2}} \right) $
$ \Rightarrow $$T = 2\cos x – 2\cos\left( {\dfrac{{8x}}{2}} \right) \cos\left( {\dfrac{{2x}}{2}} \right) $
$ \Rightarrow $$T = 2\cos x – 2\cos 4x \cos x$
Factor out the common term.
$T = 2\cos x\left( {1 - \cos4x} \right)$
Now apply the formula $\cos^{2}A + \sin^{2}A = 1$.
$T = 2\cos x\left( {\cos^{2}2x + \sin^{2}2x - \cos4x} \right)$
$ \Rightarrow $$T = 2\cos x\left( {\cos^{2}2x + \sin^{2}2x - \left( {\cos^{2}2x - \sin^{2}2x} \right)} \right)$ [Since $\cos 2A = \cos^{2}A - \sin^{2}A$]
$ \Rightarrow $$T = 2\cos x\left( {2\sin^{2}2x} \right)$
$ \Rightarrow $$T = 4\cos x{\left( {\sin2x} \right)^2}$ [ Since $\sin^{2}A = {\left( {\sin A} \right)^2}$]

Apply the formula $\sin 2A = 2\sin A \cos A$.
$T = 4\cos x{\left( {2\sin x \cos x} \right)^2}$
Simplify the above equation.
$T = 4cosx\left( {4\sin^{2}x \cos^{2}x} \right)$
$ \Rightarrow $$T = 16\cos^{3}x \sin^{2}x $

Option ‘A’ is correct

Note: Trigonometric expressions can be solved by converting the sum or differences of sine or cosine functions to the product of trigonometric ratios.
Trigonometric ratios of compound angles include evaluation of trigonometric sum or trigonometric difference of two or more angles.