0
votes
0answers
44 views

Find a short formula for $\sin x+\sin (x+y)+\sin (x+2y)+. . .+\sin (x+(n-1)y)$

The answer is : $$\sin(\frac {x+x+(n-1)y}{2}) \dfrac {\sin \frac{ny}{2}}{\sin \frac {y}{2}}$$ I could've written the question as: Show that..., but then people would try induction. What I did: ...
0
votes
2answers
40 views

A simpler way to prove this trigonometric identity?

The question asks to prove: $$ \frac{\tan{A}}{1-\cot{A}}+ \frac{\cot{A}}{1-\tan{A}}=\sec{A}\csc{A}+1$$ using only: $$ \sin^2{A}+\cos^2{A}=1\;\; \text{ & }\; \;\tan^2{A}+1=\sec^2{A}\;\; \text{ ...
2
votes
0answers
45 views

Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
3
votes
3answers
421 views

Prove $1 + \cot^2\theta = \csc^2\theta$

Prove the following identity: $$1 + \cot^2\theta = \csc^2\theta$$ This question is asked because I am curious to know the different ways of proving this identity depending on different ...
2
votes
4answers
203 views

Prove $1 + \tan^2\theta = \sec^2\theta$

Prove the following trigonometric identity: $$1 + \tan^2\theta = \sec^2\theta$$ I'm curious to know of the different ways of proving this depending on different characterizations of tangent and ...
5
votes
0answers
56 views

Sum of squares of cotangents (Check properly of expression)

I found exercise in "Introduction to algebra" Part I (A.I. Kostrikin) Check expression $\sum_{k=1}^n\cot^2\frac{k\pi}{2n+1}=\frac{n(2n-1)}{3}$ for $n=1,2,3,4,5$. For $n=1,2$ it is simple. ...
6
votes
11answers
851 views

How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$?

How to prove that $$\lim\limits_{x\to0}\frac{\tan x}x=1?$$ I'm looking for a method beside L'Hospital's rule.
3
votes
4answers
144 views

Prove $\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A$ by the Pythagorean theorem.

How do I use the Pythagorean Theorem to prove that $$\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A?$$
3
votes
1answer
150 views

Largest element of the set $\{ \sin{1}, \sin{2}, \sin{3}\}$

i have to find the largest element of the following set $\{ \sin{1}, \sin{2}, \sin{3}\}$. I converted every element to the first quadrant so i can use the monotony of cosine, the set becomes: ...
1
vote
2answers
220 views

Hard proof concerning the periodicity of trigonometrical functions. Is that a challenge or just trivial

i want to know if exist or if you can develop or give me ideas of a proof to show that the least number for which sine is periodic is $2\pi$ $$\neg \{\exists n\in \mathbb{R} \wedge n < 2\pi: ...
4
votes
1answer
130 views

On the differential equation $y''+y=0$

Consider the differential equation $$\frac{d^{2}y}{dx^{2}}+y=0$$ with initial conditions $y(0)=0$ and $y'(0)=1$. The solution is well known - $y=\sin(x)$. I know how to derive this solution, since the ...
10
votes
2answers
750 views

Algebraic proof of a trig matrix identity?

I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question: I have a geometric proof, but is there an elegant algebraic proof ...