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

Hint: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
\[\sin \left( {90 - \theta } \right) = \cos \theta \\
\Rightarrow \cos \left( {90 - \theta } \right) = \sin \theta \\ \]
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.