A calculus variable conversion question

Question

Express $\theta$ in terms of x if:

$$x+(1/x) = \sqrt{2}\cdot \sec (\theta)$$

A complete solution is always welcome

Answer 1

Hints:

• Square both the sides.

• Use the fact that $\sec^{2}(\theta) - 1 = \tan^{2}(\theta)$.

Final answer will be:

$$\theta = \tan^{-1}\biggl( \sqrt{\frac{1}{2}\cdot \Bigl(x^{2}+\frac{1}{x^2}\Bigr)}\biggr)$$

Answer 2

The function $f(x)=x+{1\over x}$ has a minimum value of $2$ for $x>0$ and a maximum value of $-2$ for $x<0$. Thus $|f(x)/\sqrt 2| \ge \sqrt 2\ge 1$ for all $x\ne0$, and $$ \theta=\sec^{-1}\Bigr(\, {\textstyle{1\over\sqrt2}} (x+{\textstyle{1\over x}})\,\Bigr) $$ where $\sec^{-1}$ is the inverse function for $\sec$ restricted to $[0,\pi]\setminus\{\pi/2\}$ (so $\sec^{-1}$ has domain $|x|\ge 1$).

Of course, as Andre points out in the comments, this isn't a unique solution.