<s>[INST] To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$ I used the following way but got wrong answer
$$A.M. \ge  G.M.$$
$$ \frac{\sin \theta + \csc  \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$
Squaring both sides,
\begin{equation*}
(\sin\theta + \csc\theta )^2 \ge 4 \tag{1} 
\end{equation*}
Similarly
\begin{equation*}
(\cos\theta + \sec\theta )^2 \ge 4 \tag{2}
\end{equation*}
Adding equation (1)  and (2)
\begin{equation*}
(\sin\theta + \csc\theta )^2+(\cos\theta + \sec\theta )^2 \ge 8
\end{equation*}
What is wrong?
 [/INST] $\begin{align}(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 & =5+\sec^2 x+\csc^2 x\\&=5+1+\tan^2 x+1+\cot^2x\\&=7+\tan^2x+\cot^2x\\&\geq9\end{align}$ 
</s>