Question

Simplify \[\dfrac{1}{4}\left[ {\sqrt 3 \cos {{23}^ \circ } - \sin {{23}^ \circ }} \right]\]
A) \[\cos {43^ \circ }\]
B) \[\cos {7^ \circ }\]
C) \[\cos {53^ \circ }\]
D) None of these

Answer 1

Hint: Given expression has trigonometric function with angles and some values also. We will try to arrange the values also in terms of function such that we will get a trigonometric formula in terms of sum and difference. Then we will write the bracket in the original formula and find it in the options or not.

Complete step by step answer:
Given the expression is,
\[\dfrac{1}{4}\left[ {\sqrt 3 \cos {{23}^ \circ } - \sin {{23}^ \circ }} \right]\]
We will write the fraction as,
\[ = \dfrac{1}{2} \times \dfrac{1}{2}\left[ {\sqrt 3 \cos {{23}^ \circ } - \sin {{23}^ \circ }} \right]\]
Now use one half to multiply with the middle terms,
\[ = \dfrac{1}{2}\left[ {\dfrac{{\sqrt 3 }}{2}\cos {{23}^ \circ } - \dfrac{1}{2}\sin {{23}^ \circ }} \right]\]
We know that, value of \[\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}\] and \[\sin {30^ \circ } = \dfrac{1}{2}\] thus this is to be written in the brackets,
\[ = \dfrac{1}{2}\left[ {\cos {{30}^ \circ }\cos {{23}^ \circ } - \sin {{30}^ \circ }\sin {{23}^ \circ }} \right]\]
As we know this is the formula for \[\cos \left( {A + B} \right)\]
\[ = \dfrac{1}{2}\left[ {\cos \left( {{{30}^ \circ } + {{23}^ \circ }} \right)} \right]\]
On adding the angles we get,
\[ = \dfrac{1}{2}\cos {53^ \circ }\]
This is the final answer. But it is not available in the options given.
So, the correct option is (D).

Note:
Here note that the value of two different trigonometric functions but the same angle are available in the bracket that’s why we can take the bracket as the formula of cos. Also note that when the fraction is multiplied with the terms inside since the terms are in product it is valid that it is taken at the denominator of any of the functions. When there is + or – sign between the terms then we can call them two separate terms.