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1

Indeed $$ (1 - \sin \alpha + \cos \alpha)^2 = (1 - \sin \alpha)^2 + \cos^2 \alpha+ 2(1 - \sin \alpha)\cos \alpha $$ $$ = (1 - 2\sin \alpha + \sin^2\alpha + \cos^2\alpha) + 2(1 - \sin \alpha)\cos \alpha $$ $$ = 2(1 - \sin \alpha) + 2(1 - \sin \alpha)\cos \alpha = 2(1 - \sin \alpha)(1 + \cos \alpha) $$


1

You calculated that the left side is equal to $$2(1-\sin\alpha + \cos\alpha -\sin\alpha\cos\alpha).$$ Now, try to prove that the left side is equal to that as well. Try expanding $$(1-\sin\alpha)(1+\cos\alpha)$$


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They are just the distributive property $$a(b+c)=ab+ac$$ in the first one we have $a=-\sin\alpha, b= 1, c=\cos\alpha$ which comes from the terms $$2(1+\underbrace{(-\sin\alpha)}_{a}\cdot\underbrace{1}_{b}+\cos\alpha+\underbrace{(-\sin\alpha)}_{a}\cdot\underbrace{\cos\alpha}_{c})$$ The next step is the same property with $a=(1+\sin\alpha), b=1, ...


0

I think that the only answer I could give is that as long as you like what you're doing, there's no wrong way to study. If you try to change your method and by doing that you don't have any pleasure, stop ! Of course there are boring and tough periods, but if you enjoy it more than it sucks... You'll do it !


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I'll hazard an answer. For both plans and videos, notes etc... much can be found from MIT's OpenCourseWear. I would say your general method of studying is wise except you probably already realize it's a bit heavy on lectures. I also wouldn't say it's necessarily wise to just work all the problems. Whatever you do, it needs to stretch your mind. Try asking or ...


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I wrote an essay based on my experiences learning math on my own over the past few years - you can read it here.


2

Théophile's answer is the standard one, but I'd like to offer a thought related to your question, if you're familiar with first-year Calculus (or even if you remember Geometric Series if you took high school precalculus). This answer, then, is more meant to supplement Théophile's. The following two "infinitely long" series of sums is valid for all real ...


3

You might be interested in looking at Cardano's method for solving cubics. Cardano himself didn't use complex numbers, but we can. Here's what happens when we use his method to solve, say, $x^3-x=0$. First set $x=u+v$, then rearrange the result so that we have: $$u^3 + v^3 + (3uv-1)(u+v) = 0.$$ Now assume that $3uv=1$. This implies from the above equation ...


1

There are many ways to build better understanding. One way that is important is to work through many examples of concepts, and as you do so, to connect them to the theory. Just learning the theorems, and the various syllogisms etc. that connect them, is not sufficient (for most students) to yield a real understanding. Similar, just working problems in ...


0

Not an answer but more of an extended comment: I tell my students to forget the textbook at first, just try to solve the exercises/problems and then if you get stuck at a problem then go read the textbook. It is a deliberate exaggeration but the point is that reading a book or article will give one a false sense of understanding. Only when we get stuck and ...


0

(I mean this as an addition to what has already been said) I think you shouldn't be afraid to take a bit of time off if you need to, and I'd personally recommend trying that before giving up altogether. A lot of people get worn out at some point, and doing a PhD does require a lot of motivation. At some point you do have to sit down and get on with ...


3

I think what Poincare calls "certain order" can also be called the mathematical idea (behind the subject you are studying). To understand a mathematical idea, the following items are important: A mathematical idea is a dynamic creature and it continually evolves according to its applications. Take for example "continuity". It started as a notion for ...


3

In my experience learning mathematics is a lot like learning a language. You need that basic vocabulary, but in order to really have a conversation you need a deep understanding of what all the words really mean and how they fit together and interact, and all the subtleties therein. Once we are proficient at a language we no longer worry about what each ...


3

This isn't a complete answer by any means. A couple of weeks ago there was a conference based on the work of William Thurston. There were several references made to an idea he used (I believe) of having levels of understanding. When you first meet something, you can read the theorems, and get a first level of understanding. But as you come back to it, in ...


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Before starting Basic Arithmetic(Pre-Arithmetic) study the definition of mathematics/Its history/influential scientists/mathematicians in it. Then the branches of mathematics and their definitions and then learn the number system and mathematical skills and number skills and think deeply about it like why it is based on ten digit then go on learning ...



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