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 colin@flyingcoloursmaths.co.uk 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

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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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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