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?

Let $s=\sin \theta$ and $c=\cos \theta$, then
\begin{align}
  E &= \tan^2 \theta+\cot^2 \theta+\sin^2 \theta+
  \cos^2 \theta+\sec^2 \theta+\csc^2 \theta \\
  &= \frac{s^2}{c^2}+\frac{c^2}{s^2}+1+\frac{1}{c^2}+\frac{1}{s^2} \\
  &= \frac{s^4+c^4+s^2c^2+s^2+c^2}{s^2c^2} \\
  &= \frac{(s^2+c^2)^2-s^2c^2+1}{s^2c^2} \\
  &= \frac{2-s^2c^2}{s^2c^2} \\
  &= \frac{2}{\sin^2 \theta \cos^2 \theta}-1 \\
  &= \frac{8}{\sin^2 2\theta}-1 \\
  & \ge 7
\end{align}
