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The value of $\cot {{18}^{\circ }}\left( \cot {{72}^{\circ }}{{\cos }^{2}}{{22}^{\circ }}+\dfrac{1}{\tan {{72}^{\circ }}{{\sec }^{2}}{{68}^{\circ }}} \right)$ ?

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Last updated date: 13th Jul 2024
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Answer
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Hint: In the given question, we need to find the value of the given trigonometric expression using various trigonometric identities and values of trigonometric functions at various angles and hence conclude some results or expression values as asked in the question.

Complete step by step answer:
According to the given question, we want to evaluate the value of the given expression which is $\cot {{18}^{\circ }}\left( \cot {{72}^{\circ }}{{\cos }^{2}}{{22}^{\circ }}+\dfrac{1}{\tan {{72}^{\circ }}{{\sec }^{2}}{{68}^{\circ }}} \right)$. Now, in order to evaluate it firstly we will sec in terms of cosine function and tangent function in terms of cot function and now the new expression would be $\cot {{18}^{\circ }}\left( \cot {{72}^{\circ }}{{\cos }^{2}}{{22}^{\circ }}+\cot {{72}^{\circ }}{{\cos }^{2}}{{68}^{\circ }} \right)$ .
Now, further we know some transformation identities of various angles by adding or subtracting angles from $\dfrac{\pi }{2}$ .
Now, we know that $\cos \left( \dfrac{\pi }{2}-x \right)=\sin x$ and similarly we know for cot function that $\cot \left( \dfrac{\pi }{2}-x \right)=\tan x$. Now, applying these two identities in the last gained expression we get
$\begin{align}
  & \cot {{18}^{\circ }}\cot \left( \dfrac{\pi }{2}-{{18}^{\circ }} \right)\left( {{\cos }^{2}}{{22}^{\circ }}+{{\cos }^{2}}{{\left( \dfrac{\pi }{2}-22 \right)}^{\circ }} \right) \\
 & \\
\end{align}$
Now, putting the values as given in the identities before we get,
$\begin{align}
  & \cot {{18}^{\circ }}\cot \left( \dfrac{\pi }{2}-{{18}^{\circ }} \right)\left( {{\cos }^{2}}{{22}^{\circ }}+{{\cos }^{2}}{{\left( \dfrac{\pi }{2}-22 \right)}^{\circ }} \right) \\
 & \Rightarrow \cot {{18}^{\circ }}\tan {{18}^{\circ }}\left( {{\cos }^{2}}{{22}^{\circ }}+{{\sin }^{2}}{{22}^{\circ }} \right) \\
\end{align}$
Now, we know that ${{\cos }^{2}}x+{{\sin }^{2}}x=1$ , so using these in the above expression we get $1\times 1=1$ .
As we know that the tan and cot functions are reciprocal of each other.
Therefore, the value of the given expression is 1.

Note: The major mistake that we make in these types of expression is we manually start evaluating the value at the given specific angles which consumes our lot of time. Instead, what we need to remember is to apply various trigonometric functions in order to reduce the calculation mistake and calculation time.