Question

Show that:
$\cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ = \dfrac{1}{2}$

Answer 1

Hint: First take the left- hand side of the equation and try to approach the right- hand side of the equation. Express the cosine angles in another form using the sign change rule of trigonometry and then eliminate the terms. Substitute the value of the obtained trigonometric ratio to get the desired result.

Complete step by step answer:
Consider the given equation:
$\cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ = \dfrac{1}{2}$
The goal of the problem is to verify the above equation.
We first take the left- hand side of the equation and try to approach the right- hand side of the equation.
The left- hand side of the equation is given as:
Left- hand side$ = \cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ $
We know that the change in the cosine is given as:
$\cos \left( {180 + \theta } \right) = - \cos \theta $ and$\cos \left( {180 - \theta } \right) = - \cos \theta $
We can also express the left- hand side as:
Left- hand side$ = \cos 24^\circ + \cos \left( {180^\circ - 125^\circ } \right) + \cos 125^\circ + \cos \left( {180^\circ + 24^\circ } \right) + \cos 300^\circ $
Here, we know that the value:
\[\cos \left( {180^\circ - 125^\circ } \right) = - 125^\circ \]and$\cos \left( {180^\circ + 24^\circ } \right) = - \cos 24^\circ $
Substitute these values in the left- hand side of the equation, then the left- hand side becomes:
Left- hand side$ = \cos 24^\circ + \left( { - \cos 125^\circ } \right) + \cos 125^\circ + \left( { - \cos 24^\circ } \right) + \cos 300^\circ $
Left- hand side$ = \cos 24^\circ - \cos 125^\circ + \cos 125^\circ - \cos 24^\circ + \cos 300^\circ $
Left- hand side$ = \cos 300^\circ $
Now, we can express the given trigonometric term in expanded form:
Left- hand side$ = \cos \left( {270 + 30} \right)^\circ $
Left- hand side$ = \sin 30^\circ $
We know that:
$\sin 30^\circ = \dfrac{1}{2}$
Then the left- hand side is given as:
Left- hand side$ = \dfrac{1}{2}$
Left- hand side$ = $ Right- hand side
We have proved the left- hand side of the equation equal to the right- hand side of the equation.
So, the left- hand side of the equation is equal to the right- hand side of the equation.
Therefore, the required result is proved.

Note: While changing the sign of the angle, note that when we are using the change in terms of $90^\circ $ and$270^\circ $, then there is an interchange in trigonometric terms.
Interchanges in the trigonometric terms are given as:
${\text{cos}} \Leftrightarrow {\text{sin}}$
${\text{tan}} \Leftrightarrow \cot $
$\sec \Leftrightarrow {\text{cosec}}$
Therefore, the interchange$\cos \left( {270 + 30} \right)^\circ = \sin 30^\circ $.