What is the least value of $\tan^2 \theta + \cot^2 \theta + \sin^2 \theta + \cos^2 \theta + \sec^2 \theta+ \textrm{cosec}^2 \theta$? What is the least value of this expression?
$$\tan^2 \theta + \cot^2 \theta + \sin^2 \theta + \cos^2 \theta + \sec^2 \theta+ \textrm{cosec}^2 \theta$$
Will putting $\theta=45^{\circ}$ give right answer?

$$\tan^2 \theta + \cot^2 \theta + \sin^2 \theta + \cos^2 \theta + \sec^2 \theta+ \textrm{cosec}^2 \theta=3+2(\tan^2\theta+\cot^2\theta)$$
Now $\tan^2\theta+\cot^2\theta=(\cot\theta-\tan\theta)^2+2\ge?$
the equality occurs if $\cot\theta-\tan\theta=0\iff\tan^2\theta=1\iff\cos2\theta=0\iff2\theta=(2m+1)\dfrac\pi2$ where $m$ is any integer
