How do you evaluate $\cos \left( {2{{\cos }^{ - 1}}\left( {\dfrac{1}{7}} \right)} \right)$?

Hint:Here we have to find the value of the given trigonometric function $\cos \left( {2{{\cos }^{ - 1}}\left( {\dfrac{1}{7}} \right)} \right)$. We will use basic trigonometric rules and identities such as $\cos 2x = 2{\cos ^2}x - 1$ and if ${\cos ^{ - 1}}(x) = y$ then $\cos y = x$. In order to solve this question we first assume that ${\cos ^{ - 1}}\left( {\dfrac{1}{7}} \right) = \theta $ then we will proceed to get the required result.


Complete step by step solution:
The inverse trigonometric functions perform the opposite operation of the trigonometric functions such as $\sin ,\,\,\cos ,\,\,\tan ,\,\,\csc ,\,\,\sec ,\,\cot \,$. The conventional symbol to represent the inverse trigonometric functions is using arc- prefix like $\arcsin \left( x \right),\arccos \left( x \right)$ or we can represent as ${\sin ^{ - 1}}x,\,\,{\cos ^{ - 1}}x,\,\,{\tan ^{ - 1}}x$. ${\sin ^{ - 1}}x,\,\,{\cos ^{ - 1}}x,\,\,{\tan ^{ - 1}}x$ denote angles or real numbers whose $\sin $ is $x$, $\cos $ is $x$ and $\tan $ is $x$ provided that the results given are numerically smallest as possible. Where ${\sin ^{ - 1}}x,\,\,{\tan ^{ - 1}}x,\,\,\,{\sec ^{ - 1}}x$ are increasing functions and ${\cos ^{ - 1}}x\,\,,\,\,{\cot ^{ - 1}}x,\,\,\,\,{\csc ^{ - 1}}x$ are decreasing functions.
Here, we have to evaluate the value of the given trigonometric function $\cos \left( {2{{\cos }^{ - 1}}\left( {\dfrac{1}{7}} \right)} \right)$.
Let ${\cos ^{ - 1}}\left( {\dfrac{1}{7}} \right) = \theta $
Then, $\cos \theta = \dfrac{1}{7}$
Substituting ${\cos ^{ - 1}}\left( {\dfrac{1}{7}} \right) = \theta $ in the given function. we get,
$ \Rightarrow \cos \left( {2\theta } \right)$
We know that $\cos 2x = 2{\cos ^2}x - 1$
So, we have
$ \Rightarrow \cos 2\theta = 2{\cos ^2}\theta - 1$
Putting $\cos \theta = \dfrac{1}{7}$ in the above equation. We get,
$ \Rightarrow 2{\cos ^2}\theta - 1 = 2{\left( {\dfrac{1}{7}} \right)^2} - 1$
$ \Rightarrow 2 \times \dfrac{1}{{49}} - 1$
Solving the above equation. We get,
$ \Rightarrow \dfrac{2}{{49}} - 1$
Simplifying the above equation by taking L.C.M (Least Common Factor). We get,
$ \Rightarrow \dfrac{{2 - 49}}{{49}}$
$ \Rightarrow \dfrac{{ - 47}}{{49}}$
Hence the value of the function $\cos \left( {2{{\cos }^{ - 1}}\left( {\dfrac{1}{7}} \right)} \right)$ is $\dfrac{{ - 47}}{{49}}$.

Note:
In order to solve these types of problems we must know all the inverse trigonometric formulas and basic trigonometric identities. Inverse trigonometric functions are also known as anti- trigonometric functions, arcus functions and cyclometric functions. These inverse trigonometric functions help us to find out any angles with any of the trigonometric ratios and derived from the properties of trigonometric functions. These types of problems can be solved by the trigonometric formulas such as $\cos 2x = 1 - 2{\sin ^2}x$ . Likewise if the function is in $\sin x$ we can use the identities of $\sin x$.