This trig problem kept me up too late last night

Originally published at:


Let’s see how the editor works…

(Sec x - tan x)^2 =
(1/cosx - sinx/cosx)^2 =
(1 - sin x )^2/ (cos x)^2 =
(1 - sin x )(1 - sin x) / (1-sinx^2)=
(1 - sin x) (1 - sin x) / (1- sin x) (1 + sin x) =
(1 - sin x)/ (1+sin x)

About 5-10 minutes for me. I have a degree in math, but haven’t used it in years for this. Still, fun problem!


I love math.

I hate this question. I hate what the system is doing to students like your daughter.

What’s the actual purpose of this course? This question? What problem-solving abilities are being amplified? Why is this being taught instead of probability? Statistics? Decision making? Even symbolic logic?

My guess is that the purpose of this question is to ensure compliance and screen for persistence.

It makes me sad.


took me like five ten minutes i think?

  • reduce everything to terms of 1, sin x, cos x
  • replace cos^2 x in denominator with Pythagorean identity
  • factor 1 - sin^2x into (1+ sin x)(1-sin x)
  • reduce.
    roughly 45 seconds. the longest operation was “confirm that sec x = 1/cos x, because who uses secant?”

EDIT: correction: the longest amount of time was the time it took to notice the fat-fingering in my answer. But neglecting that…


Trig only started to make actual sense to me when I started making video games.


The trick is to move up to the next level of mathematics where you’re really lost, then everything before it doesn’t seem so bad. I didn’t really appreciate Taylor series until I started using them in engineering, usually in the form of “and for very small x, sin x = ~ x. Because all the rest is small and meaningless, and we’re engineers, not mathematicians.”


Yeah, if you’re used to it you look at it and recognize the factors almost by reflex, and after that it’s two lines in your head but write them out long form on paper.

IF you’re used to it.

What I taught my kids about halfway through HS trig was Euler’s exponential for for trig functions, with a couple of exercises so they were comfortable with both the e^ix=cos(x)+i sin(x) and the sin() and cos() in terms of the complex exponents. It’s slightly more work than if you have the other identities memorized, but it’s AWESOME if you either get stuck or want to check your work.

Then, when you hit calculus (and more so DEs) you can work on something other than recalling trig class. Which, as it happens, you’re more likely to remember because you learnt most of it two different ways.

Take this for what it might be worth – I’m a retired engineer who decided to reset my transcript and take a BS Physics and BS Math from the beginning because (go ahead, tell people – nobody believes it) they’re vastly more fun than the other stuff that geezers do.


I can’t. This stuff always stumped me in HS.


Heretic!! Blasphemer!!

Always remember that d/dx tan(x) = sec^2(x) because you use it a LOT in Calc 2.


@Ray_Cornwall : same approach. Fun problem!

@sethgodin : Creative problem solving. Given a set of constraints (trig identities) and those good ol’ tricks from elementary algebra (e.g., a^2 - b^2) show how to get from here to there, step by step. Mental gymnastics.

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To some degree or other I enjoyed every math course I took before and after Trig. But man I always hated Trig.


About 30 seconds. It went faster cuz I wrote c for cos x and s for sin x.

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As you see in the first comment’s solution, it’s to reinforce the recognition of the algebraic identity (1 - x*x) = (1 - x) (1 + x) in the seemingly unrelated realm of trigonometry. This will, in turn, show up in calculus repeatedly (especially when they cover integration by parts).

Stop Stealing Mathematics And Objective Thought.


Euler’s identity is also a quick way to remember/verify the double- and half-angle trigonometric identities. Which was dead useful when I was coding up a quaternion class.



Er… I mean trigonomic? goes back to pile of history books and hugs them


Now that sounds interesting. What was the immediate intended application?

Hear, hear! Down with secant, cosecant and cotangent!

I’m always doubtful as to the value of this kind of problem when it doesn’t serve a more meaningful derivation. My Russian colleagues would disagree and say that it’s much-needed technique practice. In any case, if you use this stuff day to day, the answer gurgles out in a quick cascade of neuronal firing, followed by a burst of chemical pleasure.


Solar system and exosystem orbit simulation and transformations between celestial coordinate systems.

Not because I’m weird, but because I work at a company that makes planetarium systems and I hate working with row-major matrices. The weirdness is tangential.


I agree with both, actually. AND I love the problem. The big GOTCHA is how the problem is presented. Is it a puzzle to be solved after the students are taught how to solve problems like this or is it a dreaded opportunity to fail? I owe a lifelong debt to my 7th grade math teacher who gave us all math-puzzle games, presented as games, to work on. Teamwork allowed, but you had more to do as well. I still remember one of the first: given the numbers 1, 2, 3, and 4 how high up the counting numbers can you go using each exactly once, plus a list of allowed operators.

We spent weeks trying to one-up each other on that one. Here I am almost sixty years later tempted to spend time on it again.

Today two of the kids have BS Physics, MS EE; one of them also has an MS Physics and ABD in physics. Their sister has a PhD in social psychology and teaches statistical research methods when she’s not running an epidemiology office. And we all LOVE playing math games. Because we all learned early that it’s awesome for its own sake.

So: teachers and teaching matter. Never doubt it, and have some appreciation for the good ones because there are a lot of things that a good math teacher can do that are easier and pay better than teaching math well.