Was My MathSE answer appropriate

Question

This meta-query focuses on my (user2661923) answer to the following link, Solving $3\cos x+4\sin x = \frac52$, as well as the subsequent comment that my answer triggered.

I encountered a mathSE query that someone else had already provided an answer to. I both liked the already provided answer and felt that I could not improve upon it. At the same time, I felt that there were still questions that needed to be addressed; that is, I felt that it would be beneficial to the OP to address these questions.

This purpose of this meta-query is to give mathSE reviewers the opportunity to express their opinions; in the circumstances that I encountered, was my answer to the original mathSE query appropriate?

If most of you feel that it was not appropriate, then, in future similar situations, I will not post such an answer. Alternative, if you feel that my answer was appropriate, then I will continue to post such answers.

By the way; some of the reactions to this meta-query have convinced me that both the title of the meta-query and the meta-query itself should be revised. Therefore, I've revised the meta-query + title.

I infer from some of the reactions to this meta-query, that my response to the original mathSE query would have been more appropriate as a comment rather than an answer. I agree. Despite that, I posted my response as an answer, simply because I thought it would make my response significantly more legible.

Edit

Recent comments discuss the use of a blog:
link = Community Blog FAQ

I have mixed feelings about relating it specifically to this meta-query.
On the one hand:

Questions on main usually lead to objective, clear, closed-ended answers. ...Expository articles, perhaps detailing how to approach a style of problems, ..., belong on the blog.

On the other hand:

How long does it take for an article to be published after it's written? ... the queue is about one month (re, as of 7-25-2014).

Addendum

In the underlying mathSE query, Yves Daoust' criticizing comment was upvoted to 5.
In this meta-query:
postmortes' 1st comment was upvoted to 8.
The meta-query itself was first upvoted to 2 and then downvoted back to 0.
The meta-query's pseudo-answer was downvoted.

Although some reviewers approved of my answer to the underlying mathSE query, it seems to me that most did not. I will be guided by this majority view.

Answer 1

This is a pseudo-answer intended to provide the original query, Ramanujan's answer, and then my answer. The idea is to allow reviewers of this meta-query to make an informed decision whether or not my answer was appropriate.

%%%%%%%%% (I added Ramanujan's answer etc.)

%%%%%%%%%

Still missing Ramunajan's answer

...........
Original Query

Find real solutions of
$3 \cos(x) + 4\sin(x) = \frac{5}{2}.$

...........
Ramanujan's Answer

I just rechecked my link to the original mathSE query. From my perspective, Ramanujan's answer is (still) gone.

...........
my (user2661923) subsequent Answer

I completely agree with Ramanujan's analysis. I also think his stopping point was well timed for the OP to complete the problem. I would like to share underlying ideas with the OP that will help him in future similar situations.

First of all, I think that it is worth demonstrating that
$t = \tan\left(\frac{x}{2}\right) \;\Rightarrow\; \cos(x) = \frac{1 - t^2}{t^2 + 1}.$

Use
(1) $\tan^2(\theta) + 1 = \sec^2(\theta).$
(2) $\cos^2(\theta) - \sin^2(\theta) = \cos(2\theta).$

Since $\sin^2(\theta) = 1 - \cos^2(\theta),\;$ (2) above can be re-expressed as
$2\cos^2(\theta) - 1 = \cos(2\theta).$

Letting $\frac{x}{2} = \theta,$ you have
$2\cos^2(x/2) - 1 = \cos(x).$

This implies that $\frac{\cos(x) + 1}{2} = \cos^2(x/2),$ which implies that
$\frac{1}{\cos^2(x/2)} = \frac{2}{\cos(x) + 1}.$

By assumption, $t = \tan\left(\frac{x}{2}\right).$
Using (1) above, this means that $t^2 + 1 = \frac{1}{\cos^2(x/2)}.$
This means that $t^2 + 1 = \frac{2}{\cos(x) + 1}$, which leads directly to the desired result.

...

This leads to some unanswered questions:

(a)
What is the source of this problem, a book, a class, or a contest? If not a contest, then what are the theorems or (previously) solved problems that led up to this problem?

(b)
What "baby steps" might the student take to attack such a problem? What type of exploration?

(c)
Assume that the problem is not from a contest. Also assume that the student has made a reasonable attempt to attack questions (a) and (b) above. This means that the student has meta-cheated, trying to figure out what recent concepts (i.e. tools) from his book or class are relevant, and the student has then taken small exploratory steps to attack the problem using these tools.

When posing this question on mathSE, the OP should share all of his analysis. That is, include the tools that the OP thinks might be pertinent, and give full details on what the OP has tried, and where the OP is having trouble.

(d)
Now consider Ramanujan's analysis, which leads directly to a solution. As shown near the start of my answer, the demonstration that
$t = \tan\left(\frac{x}{2}\right) \;\Rightarrow\; \cos(x) = \frac{1 - t^2}{t^2 + 1}$
involves some complicated analysis.

Further, this begs the question: How was the OP (i.e. the student) supposed to know that the substitution $t = \tan(x/2)$ would lead to a solution?

...

If this problem is from a contest, then the atmosphere is obviously "all bets are off", and it's (very) fair game that the solution requires unpredictable creativity.

To the OP: if this is the case, then you can ignore the rest of this answer.

If this problem is (instead) from a book or a class, then have you (the OP) been exposed to recent book/class problems that required a substitution like $t = \tan(x/2)$?

Further, have you been exposed to problems/theorems that utilized analysis similar to the analysis that I shared that demonstrated that
$t = \tan\left(\frac{x}{2}\right) \;\Rightarrow\; \cos(x) = \frac{1 - t^2}{t^2 + 1}.$

In other words, is the solution based on Ramanujan's analysis the solution that your teacher or book intended the student to derive? If not, then what "baby steps" was the student supposed to take that would have (eventually) solved the problem?

The point of my long-winded answer is:
(e) How should the student attack a book/class problem?
(f) How should the student present (any) problem on mathSE.