Answer
See the full explanation below.
Work Step by Step
${{\cos }^{4}}t-{{\sin }^{4}}t$
Now factorize the above expression. Apply the formulae ${{A}^{2}}-{{B}^{2}}=(A+B)(A-B)$, with $A={{\cos }^{2}}t$, and $B={{\sin }^{2}}t$ for the first numeric expression.
${{\cos }^{4}}t-{{\sin }^{4}}t=\left( {{\cos }^{2}}t+{{\sin }^{2}}t \right)\left( {{\cos }^{2}}t-{{\sin }^{2}}t \right)$
Apply the Pythagorean identity of trigonometry ${{\sin }^{2}}t+{{\cos }^{2}}t=1$. Then, the above expression can be further simplified as:
$\begin{align}
& {{\cos }^{4}}t-{{\sin }^{4}}t=1\left( {{\cos }^{2}}t-{{\sin }^{2}}t \right) \\
& ={{\cos }^{2}}t-{{\sin }^{2}}t
\end{align}$
We use the Pythagorean identity of trigonometry ${{\cos }^{2}}t=1-{{\sin }^{2}}t$, which comes out by solving ${{\sin }^{2}}t+{{\cos }^{2}}t=1$. Now, the above expression can be further simplified as:
$\begin{align}
& {{\cos }^{2}}t-{{\sin }^{2}}t=1-{{\sin }^{2}}t-{{\sin }^{2}}t \\
& =1-2{{\sin }^{2}}t
\end{align}$
Thus, the left side of the expression is equal to the right side, which is
${{\cos }^{4}}t-{{\sin }^{4}}t=1-2{{\sin }^{2}}t$.
