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 .

A nice way to simplify this problem, using nothing more than elementary trigonometry and algebra, is this:
As noted in another answer, the given expression simplifies to:
$$2\sec^2\theta +2\csc^2\theta - 1$$
Finding $\theta$ to minimize that expression is the same as finding $\theta$ to minimize 
$$\begin{align}
\sec^2\theta+\csc^2\theta &= \frac{1}{\cos^2\theta}+\frac{1}{\sin^2\theta}\\
&=\frac{\sin^2\theta+\cos^2\theta}{\sin^2\theta\cos^2\theta}\\
&=\frac{1}{\sin^2\theta\cos^2\theta},
\end{align}$$
which is the same as finding $\theta$ to maximize
$$\sin^2\theta\cos^2\theta = (\sin\theta\cos\theta)^2,$$
which is the same as finding $\theta$ to maximize
$$|\sin\theta\cos\theta| = \frac12|\sin(2\theta)|.$$
This is clearly maximized when $\sin(2\theta)=\pm 1$, which happens when $\theta$ is $45^\circ$ away from a multiple of $90^\circ$. If you plug in $\theta=\frac{\pi}{4}$, you'll find that minimum value.
