91Ó°ÊÓ

The first thing to do is rewrite \(\tan{\theta}\) and \(\cot{\theta}\) in terms of \(\sin{\theta}\) and \(\cos{\theta}\). Recall that \(\tan{\theta}=\frac{\sin{\theta}}{\cos{\theta}}\) and \(\cot{\theta}=\frac{\cos{\theta}}{\sin{\theta}}\).

Now, we replace \(\tan{\theta}\) and \(\cot{\theta}\) in the expression and expand the squared term: \[4+\left(\frac{\sin{\theta}}{\cos{\theta}}-\frac{\cos{\theta}}{\sin{\theta}}\right)^2.\] Let's expand the square: \begin{align*} \left(\frac{\sin{\theta}}{\cos{\theta}}-\frac{\cos{\theta}}{\sin{\theta}}\right)^2 &= \left(\frac{\sin^2{\theta}}{\cos^2{\theta}}-2\frac{\sin{\theta}\cos{\theta}}{\sin{\theta}\cos{\theta}}+\frac{\cos^2{\theta}}{\sin^2{\theta}}\right) \\ &= \frac{\sin^2{\theta}}{\cos^2{\theta}}+\frac{\cos^2{\theta}}{\sin^2{\theta}}-2. \end{align*} So our expression is now: \[4+\frac{\sin^2{\theta}}{\cos^2{\theta}}+\frac{\cos^2{\theta}}{\sin^2{\theta}}-2.\]

Now, let's simplify the expression and rewrite it using \(\sec^2{\theta}\) and \(\csc^2{\theta}\): \begin{align*} 4+\frac{\sin^2{\theta}}{\cos^2{\theta}}+\frac{\cos^2{\theta}}{\sin^2{\theta}}-2 &= \frac{\sin^2{\theta}}{\cos^2{\theta}}+\frac{\cos^2{\theta}}{\sin^2{\theta}}+2 \\ &= \frac{\sin^2{\theta}\sin^2{\theta}+\cos^2{\theta}\cos^2{\theta}}{\sin^2{\theta}\cos^2{\theta}} \\ &= \frac{\sin^4{\theta}+\cos^4{\theta}}{\sin^2{\theta}\cos^2{\theta}}. \end{align*} Now, we rewrite the expression using the definitions of \(\sec{\theta}\) and \(\csc{\theta}\): \[\frac{\sin^4{\theta}+\cos^4{\theta}}{\sin^2{\theta}\cos^2{\theta}} = \frac{\sin^4{\theta}}{\sin^2{\theta}\cos^2{\theta}} + \frac{\cos^4{\theta}}{\sin^2{\theta}\cos^2{\theta}}.\] Recall that \(\sec{\theta}=\frac{1}{\cos{\theta}}\) and \(\csc{\theta}=\frac{1}{\sin{\theta}}\). Squaring both sides, we have \(\sec^2{\theta}=\frac{1}{\cos^2{\theta}}\) and \(\csc^2{\theta}=\frac{1}{\sin^2{\theta}}\). Therefore, our simplified expression is: \[\sec^2{\theta}\csc^2{\theta} +\csc^2{\theta}\sec^2{\theta} = \sec^2{\theta}+\csc^{2}{\theta}.\]