Evaluate the following expression: $\dfrac{2}{3}\left( {{\cos }^{4}}30-{{\sin }^{4}}45 \right)-3\left( {{\sin }^{2}}60-{{\sec }^{2}}45 \right)+\dfrac{{{\cot }^{2}}30}{4}$

Hint:We will solve this question by replacing the value of $\cos 30=\dfrac{\sqrt{3}}{2}$ , $\sin 45=\dfrac{1}{\sqrt{2}}$ , $\sin 60=\dfrac{\sqrt{3}}{2}$, $\cot 30=\sqrt{3}$ and $\sec 45=\sqrt{2}$. And then we will solve the expression that we get after replacing the value in $\dfrac{2}{3}\left( {{\cos }^{4}}30-{{\sin }^{4}}45 \right)-3\left( {{\sin }^{2}}60-{{\sec }^{2}}45 \right)+\dfrac{{{\cot }^{2}}30}{4}$, and then we will perform some algebraic operation and simplify the given expression.


Complete step-by-step answer:
First we will write the given expression and then we will substitute the values in that expression that are known to us.
In the given expression $\dfrac{2}{3}\left( {{\cos }^{4}}30-{{\sin }^{4}}45 \right)-3\left( {{\sin }^{2}}60-{{\sec }^{2}}45 \right)+\dfrac{{{\cot }^{2}}30}{4}$, we will substitute the value $\cos 30=\dfrac{\sqrt{3}}{2}$ , $\sin 45=\dfrac{1}{\sqrt{2}}$ , $\sin 60=\dfrac{\sqrt{3}}{2}$, $\cot 30=\sqrt{3}$ and $\sec 45=\sqrt{2}$.
Hence, after substituting we get the expression as,
$\begin{align}
  & =\dfrac{2}{3}\left( {{\left( \dfrac{\sqrt{3}}{2} \right)}^{4}}-{{\left( \dfrac{1}{\sqrt{2}} \right)}^{4}} \right)-3\left( {{\left( \dfrac{\sqrt{3}}{2} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}} \right)+\dfrac{{{\left( \sqrt{3} \right)}^{2}}}{4} \\
 & =\dfrac{2}{3}\left( \dfrac{9}{16}-\dfrac{1}{4} \right)-3\left( \dfrac{3}{4}-2 \right)+\dfrac{3}{4} \\
 & =\dfrac{2}{3}\left( \dfrac{5}{16} \right)-3\left( \dfrac{-5}{4} \right)+\dfrac{3}{4} \\
 & =\dfrac{5}{24}+\dfrac{15}{4}+\dfrac{3}{4} \\
 & =\dfrac{113}{24} \\
\end{align}$
Hence, after solving the expression we get the value as $\dfrac{113}{24}$ .

Note: To solve this question we have used known values $\cos 30=\dfrac{\sqrt{3}}{2}$ , $\sin 45=\dfrac{1}{\sqrt{2}}$ , $\sin 60=\dfrac{\sqrt{3}}{2}$, $\cot 30=\sqrt{3}$ and $\sec 45=\sqrt{2}$ , and that values must be known to student also. One can also use some trigonometric and algebraic formula to simplify the given expression first and the substitute the value in that simplified expression to get the final answer, in both the case the answer will be same but in the second method we have reduced the amount of calculation so it can be a better option to reduce the chance of making mistakes.