<s>[INST] Find the minimum value of $\sin^{2} \theta +\cos^{2} \theta+\sec^{2} \theta+\csc^{2} \theta+\tan^{2} \theta+\cot^{2} \theta$ 
Find the minimum value of
$\sin^{2} \theta +\cos^{2} \theta+\sec^{2} \theta+\csc^{2} \theta+\tan^{2} \theta+\cot^{2} \theta$

$a.)\ 1 \ \ \ \ \ \ \ \ \ \ \ \ b.)\ 3 \\
c.)\ 5 \ \ \ \ \ \ \ \ \ \ \ \ d.)\ 7 $
$\sin^{2} \theta +\cos^{2} \theta+\sec^{2} \theta+\csc^{2} \theta+\tan^{2} \theta+\cot^{2} \theta \\ 
=\sin^{2} \theta +\dfrac{1}{\sin^{2} \theta }+\cos^{2} \theta+\dfrac{1}{\cos^{2} \theta }+\tan^{2} \theta+\dfrac{1}{\tan^{2} \theta } \\
\color{blue}{\text{By using the AM-GM inequlity}} \\
\color{blue}{x+\dfrac{1}{x} \geq 2} \\ 
=2+2+2=6 $
Which is not in options.
But I am not sure if I can use that $ AM-GM$ inequality in this case.
I look for a short and simple way .
I have studied maths upnto $12$th grade .
 [/INST] Just using algebra$$y=\sin^{2} (\theta) +\cos^{2} (\theta)+\sec^{2} (\theta)+\csc^{2} (\theta)+\tan^{2} (\theta)+\cot^{2} (\theta)=2 \csc ^2(\theta )+2 \sec ^2(\theta )-1$$ Taking derivatives and simplifying $$y'=4 \tan (\theta) \sec ^2(\theta)-4 \cot (\theta) \csc ^2(\theta)=-32 \cot (2 \theta) \csc ^2(2 \theta)$$ $$y''=64 \csc ^4(2 \theta)+128 \cot ^2(2 \theta) \csc ^2(2 \theta)=64 (\cos (4 \theta)+2) \csc ^4(2 \theta)$$ The second derivative is always positive, so we can only find minimum values if $y'=0$ and this happens for $\theta=\pm\frac \pi 4$ and $\theta=\pm\frac {3\pi} 4$. For these values, $y=7$.
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