Answer
See the explanation below.
Work Step by Step
${{\left( 3\cos \theta -4\sin \theta \right)}^{2}}+{{\left( 4\cos \theta +3\sin \theta \right)}^{2}}$
Now, factorize the above expression. By using the formulae ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ and ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ ,with $A=3\cos \theta $ and $B=4\sin \theta $ , and $A=4\cos \theta $ and $B=3\sin \theta $ , respectively, for the numeric expression.
$\begin{align}
& {{\left( 3\cos \theta -4\sin \theta \right)}^{2}}+{{\left( 4\cos \theta +3\sin \theta \right)}^{2}}={{\left( 3\cos \theta \right)}^{2}}-2.3\cos \theta .4\sin \theta +{{\left( 4sin\theta \right)}^{2}}+{{\left( 4\cos \theta \right)}^{2}} \\
& +2.4\cos \theta .3\sin \theta +{{\left( 3sin\theta \right)}^{2}} \\
& =9{{\cos }^{2}}\theta -24\cos \theta \sin \theta +16{{\sin }^{2}}\theta +16{{\cos }^{2}}\theta \\
& +24\cos \theta \sin \theta +9{{\sin }^{2}}\theta
\end{align}$
Regroup the terms in the above expression and factor and simply it:
$\begin{align}
& 9{{\cos }^{2}}\theta -24\cos \theta \sin \theta +16{{\sin }^{2}}\theta +16{{\cos }^{2}}\theta \\
& +24\cos \theta \sin \theta +9{{\sin }^{2}}\theta \\
& =9{{\cos }^{2}}\theta +9{{\sin }^{2}}\theta +16{{\sin }^{2}}\theta +16{{\cos }^{2}}\theta \\
& =9\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)+16\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)
\end{align}$
By using the Pythagorean identity of trigonometry ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ , the above expression can be further simplified as:
$\begin{align}
& 9\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)+16\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)=9\left( 1 \right)+16\left( 1 \right) \\
& =9+16 \\
& =25
\end{align}$
Thus, the left side of the expression is equal to the right side, which is
${{\left( 3\cos \theta -4\sin \theta \right)}^{2}}+{{\left( 4\cos \theta +3\sin \theta \right)}^{2}}=25$.
