Is there an easier way to eliminate a variable from equations?

Question

I was wondering if theres an easier way to do these types of problems that ask to eliminate a variable, since these types of problems are asked in timed tests, and I really can't afford to spend 10 minutes on each question.

There are certain trigonometric equations like: $$m=\tan x +\sin x$$ $$n=\tan x -\sin x$$ The question asks to eliminate x from the equation

My approach with these kind of problems are to just mess around with $m$ and $n$ until I get the trigonometric identity $\sin^2 x +\cos^2 x= 1$

These kind of problems are also there in locus of a point about certain conditions, like :

A rod of length l slides with its ends on two perpendicular lines. Find the locus of its midpoint.

Answer 1

$\frac {m+n} 2=\tan x$ and $\frac {m-n} 2=\sin x$. Now use the following: $1+\tan ^{2}x=\frac 1 {1-\sin ^{2}x}$.

Answer 2

Solve the simultaneous equations for $$\tan x,\sin x$$

Now use $$\dfrac1{\sin^2x}-\dfrac1{\tan^2x}=\cdots=1$$