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

Last updated date: 13th Jul 2024
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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.

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}
\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 $1\times 1=1$ .