Question

The value of $\cos {12^ \circ } + \cos {84^ \circ } + \cos {156^ \circ } + \cos {132^ \circ }$ is
A. $\dfrac{1}{2}$
B. $1$
C. $\dfrac{{ - 1}}{2}$
D. $\dfrac{1}{8}$

Answer 1

Here in this question, we have to find the exact value of a given trigonometric function. For this first we have to solve the given expression using a Sum to Product Formula of trigonometry i.e., $\cos x + \cos y = 2\cos \left( {\dfrac{{x + y}}{2}} \right)\cos \left( {\dfrac{{x - y}}{2}} \right)$ and by using the complementary angle ${90^ \circ }$ and further simply using the standard values of angles of trigonometric ratios to get the required solution.

Complete step by step answer:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle; the sine, cosine, tangent, cotangent, secant, or cosecant known as trigonometric function Also called circular function. Consider the given question:
$\cos {12^ \circ } + \cos {84^ \circ } + \cos {156^ \circ } + \cos {132^ \circ }$
On rearranging, we can written the above given equation as
$\Rightarrow \,\,\,\,\left( {\cos {{132}^ \circ } + \cos {{12}^ \circ }} \right) + \left( {\cos {{156}^ \circ } + \cos {{84}^ \circ }} \right)$ -------(1)
Now, apply the sum to product formula of trigonometry i.e.,
$\cos x + \cos y = 2\cos \left( {\dfrac{{x + y}}{2}} \right)\cos \left( {\dfrac{{x - y}}{2}} \right)$
Then equation (1), becomes
$\Rightarrow \,\,\,2\cos \left( {\dfrac{{132 + 12}}{2}} \right)\cos \left( {\dfrac{{132 - 12}}{2}} \right) + 2\cos \left( {\dfrac{{156 + 84}}{2}} \right)\cos \left( {\dfrac{{156 - 84}}{2}} \right)$
$\Rightarrow \,\,\,2\cos \left( {\dfrac{{144}}{2}} \right)\cos \left( {\dfrac{{120}}{2}} \right) + 2\cos \left( {\dfrac{{240}}{2}} \right)\cos \left( {\dfrac{{72}}{2}} \right)$

On simplification, we get
$\Rightarrow \,\,\,2\cos \left( {{{72}^ \circ }} \right)\cos \left( {{{60}^ \circ }} \right) + 2\cos \left( {{{120}^ \circ }} \right)\cos \left( {{{36}^ \circ }} \right)$ -----(2)
As we know, from the standard angles table of trigonometric ratios the value of $\cos {60^ \circ } = \dfrac{1}{2}$ and $\cos {120^ \circ } = - \dfrac{1}{2}$.
On substituting the values in equation (2), then
$\Rightarrow \,\,\,2\cos \left( {{{72}^ \circ }} \right)\left( {\dfrac{1}{2}} \right) + 2\left( { - \dfrac{1}{2}} \right)\cos \left( {{{36}^ \circ }} \right)$
On simplification, we get
$\Rightarrow \,\,\,\cos \left( {{{72}^ \circ }} \right) - \cos \left( {{{36}^ \circ }} \right)$ ----(3)
$\cos \left( {{{72}^ \circ }} \right)$ can be written in difference of ${90^ \circ }$ is $\cos \left( {{{72}^ \circ }} \right) = \cos \left( {90 - {{18}^ \circ }} \right)$, then equation (3) becomes
$\Rightarrow \,\,\,\cos \left( {90 - 18} \right) - \cos \left( {{{36}^ \circ }} \right)$ ----(4)

Let us by the complementary angles of trigonometric ratios:
The angle can be written as

\Rightarrow \cos \left( {90 - \theta } \right) = \sin \theta \\\ $$ On substituting in equation (4), we have $$ \Rightarrow \,\,\,\sin {18^ \circ } - \cos {36^ \circ }$$ ------(5) By the standard calculator of trigonometric ratios the value of $$\sin {18^ \circ } = \dfrac{{\sqrt 5 - 1}}{4}$$ and $$\sin {36^ \circ } = \dfrac{{\sqrt 5 + 1}}{4}$$. On substituting the values in equation (5), we have $$ \Rightarrow \,\,\,\dfrac{{\sqrt 5 - 1}}{4} - \left( {\dfrac{{\sqrt 5 + 1}}{4}} \right)$$ $$ \Rightarrow \,\,\,\dfrac{{\sqrt 5 - 1 - \sqrt 5 - 1}}{4}$$ On simplification, we get $$ \Rightarrow \,\,\,\dfrac{{ - 2}}{4}$$ Divide, both numerator and denominator by 2, then we get $$\therefore \,\,\,\dfrac{{ - 1}}{2}$$ Hence, the required solution is $$\dfrac{{ - 1}}{2}$$. **Therefore, option C is the correct answer.** **Note:** When solving the trigonometry-based questions, we have to know the definitions and table of standard angles of all six trigonometric ratios. Remember, when the sum of two angles is $${90^ \circ }$$, then the angles are known as complementary angles at that time the ratios will change like $$\sin \leftrightarrow \cos $$, $$\sec \leftrightarrow cosec$$ and $$\tan \leftrightarrow \cot $$ then should know the some basic formulas of trigonometry like identities, double and half angle formulas, Product to Sum Formulas and Sum to Product Formulas.