Answer
The expression on the left-side is equal to the expression on the right-side.
Work Step by Step
Let us consider the given expression on the left side ${{\cos }^{4}}t-{{\sin }^{4}}t$ can be further simplified as:
${{\cos }^{4}}t-{{\sin }^{4}}t=\left( {{\cos }^{2}}t-{{\sin }^{2}}t \right)\left( {{\cos }^{2}}t+{{\sin }^{2}}t \right)$
As per the double angle formula, $\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta $ , and as per the Pythagorean identity, ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$. Therefore, applying the identities, we get:
$\begin{align}
& {{\cos }^{4}}t-{{\sin }^{4}}t=\left( {{\cos }^{2}}t-{{\sin }^{2}}t \right)\left( {{\cos }^{2}}t+{{\sin }^{2}}t \right) \\
& =\cos 2t.1 \\
& =\cos 2t
\end{align}$
Hence, the expression on the left-side is equal to the expression on the right-side.
