I have a question regarding tutoring a calculus student. They need to prove trig identities such as $$\frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2\sec^2x.$$ Doing this kind of problem is very tedious and time consuming. Is it really so necessary to focus on this for the learning of calculus or can this part be done more superficially?
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7$\begingroup$ Are you not comfortable explaining the required manipulations? This particular one is 3 steps. To be fair, however, this tends to appear in trig/pre-calc more than calc, itself. Your student, if in calculus already, is still in the review phase, in my opinion. $\endgroup$– JTP - Apologise to MonicaCommented Sep 24, 2019 at 2:16
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23$\begingroup$ @Burt I might reconsider whether you are really qualified to tutor someone in calculus if you do not feel confident in this sort of routine calculation. $\endgroup$– Steven GubkinCommented Sep 24, 2019 at 12:10
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1$\begingroup$ I see good answers have posted. Last night, my inclination was to share a trig manipulation that was difficult enough that teachers noted that such problems would not be on the chapter test. It's not an answer, because it doesn't really address the tie in to calc. I'm sharing as an example of one of the tougher ones I've seen. $\endgroup$– JTP - Apologise to MonicaCommented Sep 24, 2019 at 12:20
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12$\begingroup$ Basically one of the most critical skills in calculus is manipulating an integrand from an apparently obtuse term to one for which the antiderivative is more apparent. For this, trig identities are often necessary, even if there are no trig terms in the integrand. $\endgroup$– BridgeburnersCommented Sep 24, 2019 at 15:25
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5$\begingroup$ As you may have noticed, many of the answers mention solving integrals. This is in Calc II in most schools. If this was assigned in a calculus course, the teacher is reviewing, and may have thrown these into an assignment without really thinking about it. (More information would be useful: Is this student in a course at a university, community college, or high school? Was this one of many problems in a textbook?) $\endgroup$– Sue VanHattum ♦Commented Sep 24, 2019 at 19:14
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9 Answers
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The specific identity \begin{equation}\tag{A} \tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x} \end{equation} as such is probably not often encountered, but simplifications akin to \begin{equation}\tag{B} \tfrac{1}{1-t} + \tfrac{1}{1 + t} = \tfrac{2}{1 - t^{2}} \end{equation} occur frequently. For example, integration via partial fractions requires undoing such a simplification, and this manipulation is impossible to understand for someone who does not understand the forward operation being undone. In the same spirit, seeing the formal similarity between (A) and (B) is relevant when it comes to making changes of variables in integrals.
Understanding that the identity (A) on the one hand involves the general algebraic identity (B) and on the other hand uses the trigonometric identity $\sin^{2}{x} + \cos^{2}{x} = 1$, and understanding how to separate these two statements, is useful for developing the sort of calculational skills that are generally necessary for making progress in calculus.
In more mercantile terms, experience teaching calculus suggests that students who cannot make manipulations such as (A) are unlikely to pass a university calculus course.
Finally, characterizing (A) as very tedious and time consuming seems to me simply wrong, as it is neither.
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Due to low enrollment, my AP Calc class was filled with the students who otherwise would have taken Pre-Calc this year. So you can imagine that "How much do you really need to know to see the bigger picture in calculus?" has been on my mind lately.
Here's where my thoughts have fleshed out in regards to trig so far.
- Periodic behavior is widespread throughout science, engineering, and the humanities, and sinusoidal functions are typically used to model periodic behavior. So simply ignoring trig is not an option.
- There is absolutely prior knowledge that students should have about trig that will be essential to understand key ideas. For instance, you cannot understand the derivative of sine without knowing that $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$. You're not going to understand how to calculate $\int\frac{dx}{\sqrt{1-x^2}}$ without understanding inverse trig functions.
- Beyond that, there really is a lot of fluff that doesn't seem to serve any real-world purpose. I'm frankly on the fence about whether secant, cosecant, and cotangent should largely go the way of versine and exsecant. Esoteric identities like the one the OP posts can have limited utility, but it seems to be largely an exercise in algebraic manipulation rather than authentic problem-solving.
In the end, of course, a student needs to know enough trig identities to be prepared for their final exam, and trig has enormous utility in the real world. But I am sympathetic to the argument that the twentieth century advanced algebra curriculum is not wholly authentic preparation for twenty-first century calculus.
answered Sep 24, 2019 at 13:38
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$\begingroup$ The secant occurs naturally, first as the derivative of tan, and second as $1 + \tan^2 x = \sec^2 x$ which is of course just another way of writing $\sin^2 x + \cos^2 x = 1$. But I agree that cot and cosec don't have much practical use. $\endgroup$ Commented Sep 25, 2019 at 8:34
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2$\begingroup$ But for "21st century calculus" students still need a solid grounding in the basics. It's depressing now many time you see students who have learned (more or less) what the notation of vector calculus looks like, but don't seem to have any idea what the symbolic manipulations actually mean - and therefore fall into rabbit holes writing things that are just nonsense, without realizing it.. $\endgroup$ Commented Sep 25, 2019 at 8:44
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$\begingroup$ @alephzero: Going through school and university in Germany, I never ever saw secant and cosecant except in an old "math for engineers" textbook on my father's bookshelf. For me the derivative of $\tan(x)$ is $\dfrac{1}{\cos^2(x)}$ or, as you say, $1+\tan^2(x)$ which is actually the most "natural" way to write it. Compare math.stackexchange.com/a/2713525/96384 with comments. $\endgroup$ Commented Oct 22, 2019 at 18:59
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From a historical perspective, knowing these identities used to be somewhat more important than now. Prior to the invention of logarithms, people who needed to do lots of sophisticated calculations [esp. astronomers] resorted to a technique know as 'prostapharesis'. This involved combining certain trigonometric identities to produce equations (e.g) having a product of trig functions on one side and a sum or difference of trig functions on the other. This allowed people to transform a multiplication into an addition or subtraction, like logarithms do (but in a somewhat more cumbersome manner).
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There are many calculus textbooks that use no trig. They may be called "Calculus for Business" or "For Biology" or "For Social Science".
random example:
Of course math, physical science, and engineering, definitely use parts of calculus connected with trig functions. I would have thought that Business would be interested in cyclic phenomena, but what do I know?
answered Sep 24, 2019 at 12:47
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2$\begingroup$ Business isn't interested in cyclic phenomena best handled by trig functions, it's interested in periodic phenomena (Christmas buying season, summer tourist season, etc.) that are best handled by numeric integration/differentiation. That said, your textbook probably covers things involving simple trig functions, eg. $\int \sin x$. $\endgroup$– MarkCommented Sep 29, 2019 at 20:40
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Algebraic skills to do manipulations in general are important. And this one is not that hard. Good practice. Get dirty and do it.
There's some applications when you get to trig subs of quadratic radicals and the like.
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Trigonometric substitutions are useful for solving many integrals in closed form and learning how to solve integrals is a major part of most calculus courses. Often more than half of university-level "Calculus II" is concerned with integration techniques. Without trigonometric identities, it may not be obvious how to solve $\int \left( \frac{1}{1- \sin x} + \frac{1}{1+\sin x} \right)~\mathrm{d}x$. However, $\int \sec^2 x~\mathrm{d}x$ is included in many integration tables and happens to be simply $\tan x + C$.
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3$\begingroup$ How often do you encounter $\int \left( \frac{1}{1- \sin x} + \frac{1}{1+\sin x} \right)~\mathrm{d}x$ outside of Calculus II? Becoming familiar with $sin^2x + cos^2x = 1$ is worthwhile because switching between sines and cosines comes up all the time, but the more exotic trig identities? $\endgroup$– MarkCommented Sep 24, 2019 at 23:29
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2$\begingroup$ If you teach that $\sin^2 x + \cos^2 x = 1$ is just Pythagoras's theorem, that alone would make it "worthwhile" to show the connection between trig and geometry. $\endgroup$ Commented Sep 25, 2019 at 8:39
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2$\begingroup$ @Mark The problem doesn’t involve any trig identities beyond $\cos^2(x) + \sin^2(x)=1$. The other manipulation is simply $(x+a)(x-a) = x^2-a^2$, which is a common substitution in real problems. If you feel like $\sec^2(x)$ is exotic, you can call it $1/\cos^2(x)$. Whatever you call it, it might be important to recognize as the derivative of $\tan(x)$. $\endgroup$ Commented Sep 25, 2019 at 21:54
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The core of concepts of calculus: that is Differentiation and Integration can be defined rigorously and intuitively without any reference to trigonometry.
To be functionally able to use these tools well also requires a very strong command over algebra which other posters have commented.
Trigonometry is in some sense a “nice to know” since it lets you apply calculus to many more real life problems that would otherwise be inaccessible but care should be taken in my opinion to separate it from the core ideas.
On a personal note: I learned Euler’s formula before I learned the double angle formula or the angle sum formulas due to rather unusual educational circumstances. I was still able to do fine in all my classes as I could use these tools to derive and confirm whatever trig I needed on the fly and I essentially learned Trig very passively this way but it was never a conceptual problem. The fact that this worked and despite popular belief I was never “crushed” by AP or High School or College or Grad Level courses for my lack of traditional trig foundation is I think supporting evidence for what I outlined above.
Trig is a VERY IMPORTANT tool that lets us apply the math. It’s not particularly necessary or useful to get to the foundations of calculus, or understand where and when calculus is necessary.
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Once the student figures out the process - start from one side of what is to be proven, use known formulas to get the other side, done (or start from both sides or do some other variation) - the rest is getting intuition about which formula to use in the process and when. I guess this intuition might be useful if they need to evaluate tricky integrals analytically, or if they do something very trigonometric later in their life.
If getting good scores from exams is important to the student, then the importance of trigonometry depends on the instructor and the syllabus. We are probably ill advised to make guesses on those, especially since there is no country specified.
For what it is worth, I have never been taught or needed any trigonometric functions but sin, cos, tan and their inverses.
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IMO knowing basic identities such as $\sin^2x + \cos^2x=1$ should be as fundamental as knowing $12\times 12 = 144$ (without needing to reach for a calculator). That is not to say that students should know all the identities by rote, but they should at least recognise the identities.
Categorising "prove trig identities such as $\frac{1}{1−\sin x}+\frac{1}{1+\sin x}=2\sec^2x$ as very tedious and time consuming" is quite the overstatement.
Basic algebra allows you to reformulate $$\frac{1}{1−\sin x}+\frac{1}{1+\sin x}=2\sec^2x$$ as $$\frac{1+\sin x}{1−\sin^2x}+\frac{1-\sin x}{1-\sin^2x}=2\sec^2x$$ which gives $$\frac{2}{\cos^2x}=2\sec^2x$$
Along with $\sin^2x+\cos^2x$, they should also then know, or at least recognise,
$\sec^2x=\frac{1}{\cos^2x}$ and $\csc^2x=\frac{1}{\sin^2x}$.
QED.
