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!

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:

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

There’s also a factor of 2 on top:

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

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