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