Ask Uncle Colin: Trigonometric craziness

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can.

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) \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


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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