<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] $$\sin^2\theta+\cos^2\theta+2+2+\csc^2\theta+\sec^2\theta$$
$$=5+\frac1{\sin^2\theta\cos^2\theta}=5+\frac4{(\sin2\theta)^2}=5+4\csc^22\theta$$
Now, $\csc^22\theta=1+\cot^22\theta\ge1$ for real $\theta$
The equality occurs if $\csc^22\theta=1\iff\sin^22\theta=1$
$\iff\cos2\theta=0\implies2\theta=(2n+1)\dfrac\pi2$ where $n$ is any integer
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