Ask Uncle Colin: Trigonometric craziness

Dear Uncle Colin,

My friend claims that $\frac{2-\frac{2\sin (x)}{\cos (x)}}{\sin (x)-\cos (x)}\equiv -2\sec (x)$. I think she’s crazy. What do you think?

-- I Don’t Even Need Trigonometry, I Teach Yoga

Hi, IDENTITY – even yoga teachers need trigonometry, though!

Well, there’s one way to find out if your friend is correct about this: work through the sums!

As usual, the first thing to do is to make the ugliest thing less ugly: here, that’s the fraction on the top of the left hand side. I’m going to multiply the fraction, top and bottom, by $\cos (x)$ to get:

$\frac{2\cos (x)-2\sin (x)}{\cos (x)(\sin (x)-\cos (x))}$

There’s also a factor of 2 on top:

$\frac{2(\cos (x)-\sin (x))}{\cos (x)(\sin (x)-\cos (x))}$

Meanwhile, $\frac{\cos (x)-\sin (x)}{\sin (x)-\cos (x)}=-1$, so the fraction is:

$\frac{-2}{\cos (x)}=-2\sec (x)$, as your friend says.

As for whether your friend is crazy, I’m not qualified to say.

-- Uncle Colin