Question

The value of \[{\cos ^2}\left( {\dfrac{\pi }{{12}}} \right) + {\cos ^2}\left( {\dfrac{\pi }{4}} \right) + {\cos ^2}\left( {\dfrac{{5\pi }}{{12}}} \right)\] is
1 \[\dfrac{3}{2}\]
2 \[\dfrac{2}{3}\]
3 \[\dfrac{{\left( {3 + \sqrt 3 } \right)}}{2}\]
4 \[\dfrac{2}{{\left( {3 + \sqrt 3 } \right)}}\]

Answer 1

Hint: We know that an equation involving one or more trigonometric ratios of unknown angles is called a trigonometric equation. To evaluate the given trigonometric function, the equation consists of cos functions, as we know that the value of \[{\cos ^2}\left( {\dfrac{\pi }{{12}}} \right)\], cannot be directly substituted hence, we need to simplify the value of \[{\cos ^2}\left( {\dfrac{\pi }{{12}}} \right)\], then substitute it in the given trigonometric function to get the value.

Complete step by step answer:
Let us write the given function:
\[{\cos ^2}\left( {\dfrac{\pi }{{12}}} \right) + {\cos ^2}\left( {\dfrac{\pi }{4}} \right) + {\cos ^2}\left( {\dfrac{{5\pi }}{{12}}} \right)\]
In which we need to find the value of this function, hence let us simplify the \[{\cos ^2}\left( {\dfrac{\pi }{{12}}} \right)\], in simplified way as:
\[{\cos ^2}\left( {\dfrac{\pi }{{12}}} \right) = {\left[ {\cos \left( {\dfrac{\pi }{2} - \dfrac{{5\pi }}{{12}}} \right)} \right]^2}\]
\[ \Rightarrow {\cos ^2}\left( {\dfrac{\pi }{{12}}} \right) = {\left( {\sin \dfrac{{5\pi }}{{12}}} \right)^2}\]
As we are given with the expression, let us substitute and evaluate the function i.e.,
\[{\cos ^2}\left( {\dfrac{\pi }{{12}}} \right) + {\cos ^2}\left( {\dfrac{\pi }{4}} \right) + {\cos ^2}\left( {\dfrac{{5\pi }}{{12}}} \right)\]
\[ = {\sin ^2}\left( {\dfrac{{5\pi }}{{12}}} \right) + {\cos ^2}\left( {\dfrac{{5\pi }}{{12}}} \right) + {\cos ^2}\left( {\dfrac{\pi }{4}} \right)\]
Simplifying the terms, we get:
\[ = 1 + {\left( {\dfrac{1}{{\sqrt 2 }}} \right)^2}\]
\[ = 1 + \left( {\dfrac{1}{2}} \right)\]
\[ = \dfrac{3}{2}\]

So, the correct answer is “Option 1”.

Note: The key point to evaluate any trigonometric function is that we must know all the basic trigonometric functions and their relation. As in the given equation consists of cos functions, hence we must know all the trigonometric identities with respect to the function.