For questions regarding identities in algebraic structures, including the construction, composition, and interpretation thereof.

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3
votes
2answers
61 views

Prove that: $\frac{2\pi i}{(1 - e^{2i\pi/n})\prod_{k=0, k \neq 1}^{n-1} (e^{i\pi/n} - e^{i(2k-1)\pi/n})} = \frac{\pi/n}{\sin(\pi/n)}$

I am trying to find $\int_0^{\infty} \frac{dx}{1 + x^n}$ using contour integration. I did the computation by taking the contour $[0,R] \cup \gamma_R \cup [R e^{2i\pi/n}, 0]$, with $\gamma_R$ the arc ...
0
votes
1answer
40 views

A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
3
votes
1answer
81 views

Show that $(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6$ for $a^2 + b^2 + c^2 + d^2 = 1$.

For $a, b, c, d \in \Bbb R$ such that $a^2 + b^2 + c^2 + d^2 = 1$, show that $$(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6.$$ The answer uses the mysterious identity ...
3
votes
1answer
117 views

Factors of the numbers of the form $a^2+nb^2$

Let $N=a^2+nb^2$ with $\gcd(a,b) =1$ and $n \in \mathbb{Z^+}$. If $N=xy$ where $x$ and $y$ are relatively prime numbers, in what condition can $x$ and $y$ be also written in the same form as $N$ ...
0
votes
1answer
28 views

Verify $\frac{\cot x -\tan x}{\cos x + \sin x}=\frac{\cos x - \sin x}{\sin x \cos x}$

Verify $$\frac{\cot x -\tan x}{\cos x + \sin x}=\frac{\cos x - \sin x}{\sin x \cos x}$$ After several tries I cannot find a concrete way of proving/verifying this. Any help/hints?
1
vote
1answer
64 views

Can composite numbers be uniquely written as a sum of two squares?

Let $X = a^2 +b^2$ where all the terms are positive integers and $X$ is a composite number and $\gcd(a,b)=1$ . Do there exist positive integers $c$ and $d$ other than $a$ and $b$ such that $X = ...
0
votes
1answer
70 views

Formal Proof that $f^{-1} \circ f = id_x \ , \forall f$

Given $f$ as an invertible function with domain $X$ and codomain $Y$, then we can say $$f^{-1}(f(x)) = x $$ Or since they are both logically equivalent $$ f(f^{-1}(x)) = x $$ This can also be ...
4
votes
1answer
123 views

For which polynomials $f$ is the subset {$f(x):x∈ℤ$} of $ℤ$ closed under multiplication?

You surely know about the Brahmagupta–Fibonacci identity, $$(a_1^2 + b_1^2)(a_2^2 + b_2^2) = (a_1a_2 \pm b_1b_2)^2 + (a_1b_2 \mp a_2b_1)^2$$ which tells us that the product of two numbers, each of ...
2
votes
2answers
31 views

Multiple Angle formulas, alternate forms

Relatively simple question, that might not be simple to answer: I have noticed that there are ways of expressing every double angle formula of a given trigonometric function using only that function ...
0
votes
1answer
20 views

Algebraic matrix manipulation

I've been under the impression that matrices cannot be treated like normal algebra. This is a fundamental concept, of course, and I've known it for some time. Generally such a notion would imply ...
0
votes
0answers
62 views

Did the book make a mistake in the identity$\frac{1}{\cos 2x+\tan x} = \sin 2x$?

EDIT: OP says that they misread the text from which this question was drawn. It actually said $$\frac{1}{\cot(2x) + \tan(x)} = \sin(2x)$$ where $\cos$ has been replaced by $\cot$; that identity is ...
-1
votes
2answers
56 views

Help me simplify: $\cos(−\theta) + \tan(−\theta) \sin(−\theta)$ [closed]

Simplify $$\cos(−\theta) + \tan(−\theta) \sin(−\theta)$$ to one term with no negative thetas.
4
votes
2answers
66 views

If $x+y+z=0$, prove that $\frac{x^2}{2x^2+yz}+\frac{y^2}{2y^2+zx}+\frac{z^2}{2z^2+xy}=1$

A problem in my homework had asked me: When $x+y+z=0$, evaluate$$\frac{x^2}{2x^2+yz}+\frac{y^2}{2y^2+zx}+\frac{z^2}{2z^2+xy}$$ Without too much difficulty, one can see that the value should be ...
1
vote
2answers
55 views

How to derive identities [duplicate]

For example: $(a+b)^2 = a^2 + b^2 + 2ab$ $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$ So my doubt regarding these identities are why does the identity differ when the power is changed and is there any ...
8
votes
1answer
62 views

How many points to prove a trigonometric identity?

I am taking a trigonometric identity from another post, arbitrarily. $$\frac{2\sec\theta +3\tan\theta+5\sin\theta-7\cos\theta+5}{2\tan\theta ...
-1
votes
3answers
28 views

How to prove this identity? (Trigonometry, reduction)

How to prove this identity? $$\frac{\sin(\pi+\alpha)}{\sin(\frac{3\pi}{2}+\alpha)} + \frac{\cos(\alpha-\pi)}{\cos(\frac{\pi}{2}+\alpha)+1} = \frac{1}{\cos\alpha}$$ I took it from exercise about ...
0
votes
1answer
29 views

Understanding this solution for a trigonometric identity of $\tan2 \theta$

I require help in the area of trigonometry in proving an identity. I am to prove that the left hand side is equal to $\tan2 \theta$. I understand up until the second step in this calculation ...
4
votes
1answer
56 views

Can $F_n^2-F_m^2$ be factored as a product of Fibonacci or Lucas numbers when $n-m$ is odd?

The Fibonacci and Lucas numbers are defined for all integers $n$ by the recurrence relations $$F_n=F_{n-1}+F_{n-2}\text{ where }F_1=1\text{ and }F_2=1,$$ $$L_n=L_{n-1}+L_{n-2}\text{ where }L_1=1\text{ ...
0
votes
2answers
46 views

Prove that $F_nF_{n+1}=\frac{1}{4}(F_{n+2}^2-F_{n-1}^2)$

The Fibonacci and Lucasnumbers are defined for all integers $n$ by the recurrence relations $$F_n=F_{n-1}+F_{n-2}\text{ where }F_1=1\text{ and }F_2=1,$$ $$L_n=L_{n-1}+L_{n-2}\text{ where }L_1=1\text{ ...
0
votes
4answers
85 views

Establish the identity of $\cos(\pi - \theta) = - \cos(\theta)$ [closed]

I need to establish the identity but am not sure how: $$\cos(\pi - \theta) = - \cos(\theta).$$
1
vote
1answer
54 views

Identity morphism requirement in categories

In order to verify a category, you need to show that the class of morphisms respect associativity and contains an identity morphism. I'm looking for a class of morphisms that doesn't contain an ...
4
votes
2answers
41 views

Trouble understanding how this identity is derived: $\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$

$$\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$$ The $-a-1$ is throwing me off. Can anyone help me understand this identity. I have tried letting $m=-a-1$ and then applying the binomial ...
1
vote
4answers
54 views

Using the usual notation for a triangle write $sin^2A$ in terms of the sides a, b and c.

This is an A-level trigonometric problem. Using the usual notation for a triangle write $sin^2A$ in terms of the sides a, b and c. Answer: $$\frac{(a+b-c)(a-b+c)(a+b+c)(-a+b+c)}{4b^2c^2}$$ The last ...
1
vote
1answer
46 views

An alternative way to compute $a^n+b^n+c^n+d^n$

I was doing the following problem: given $a, b, c, d \in \mathbb{R}$ and $a+b+c+d=1$, $a^2+b^2+c^2+d^2=2$, $a^3+b^3+c^3+d^3=3$ and $a^4+b^4+c^4+d^4=4$. Find $a^n+b^n+c^n+d^n$ (I am looking for a ...
2
votes
1answer
29 views

Linear relations among wedge products.

Let $v_1,v_2,v_3,v_4$ be $4$ vectors in a two dimensional space $V$. Then one can work out by hand that: $$(v_1\wedge v_2)(v_3\wedge v_4) + (v_1\wedge v_3)(v_4\wedge v_2) + (v_1\wedge v_4)(v_2\wedge ...
2
votes
1answer
47 views

Algebraic identity involving powers of twin primes

Yesterday, I verified that, if $a$,$b$ and $c$ are real numbers such that $a+b+c=0$, then $$\frac{a^5+b^5+c^5}{5}=\frac{a^3+b^3+c^3}{3}\cdot\frac{a^2+b^2+c^2}{2}$$ and ...
0
votes
1answer
38 views

What floor function identity makes this true?

I know that the graph of these two functions is the same: $$(-1)^{\lfloor x\rfloor} = -2\lfloor x\rfloor + 4\left\lfloor\frac {\lfloor x\rfloor}2\right\rfloor + 1$$ Both of them interchange sign at ...
1
vote
0answers
52 views

Any linear way to express this expression

Do you think that there is a linear way to express this, using maybe a characteristic of this system that I cannot see. This is a set of equations involving exactly all sets of $(x_i, y_j, y_k)$ or ...
0
votes
1answer
33 views

Express a variable as a function of another [closed]

If we have this formula : $a/(1-$$1\over x$$-$$1\over y$$) = 1/(1-$$1\over xb$$-$$1\over yb$$)$ Is it possible to express $a$ as a function of $b$, independently of $x$ or $y$ ? Thank you
5
votes
2answers
57 views

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$? Because of the nature of the square root function, its derivative monotonically decreases. so ...
0
votes
3answers
27 views

Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
1
vote
1answer
45 views

Trigonometric solution for $\int_{0}^{2 \pi} \sin^n (x) \cos^m (x) dx $?

At home I came across the exercise and had to compute: $\int_{0}^{2\pi} \sin^n (x) \cos^m (x) dx $ with $m$, $n \in \mathbb{N} $ My current set of tools for solving problems of that kind is rather ...
15
votes
3answers
160 views

Is there no formula for $\cos(x^2)$?

I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance ...
1
vote
2answers
38 views

Is there a name for this double summation identity? What is the shortest way to illustrate that it holds?

Say I have the following the expression: $$\sum\limits_{j=0}^{i-1} \sum\limits_{u=0}^{j} g(u)$$ By enumeration, it is easy to see that: in the case when $j=0$ we have $$\sum\limits_{u=0}^{j} g(u) ...
3
votes
1answer
590 views

Does this matrix identity hold?

For invertible matrices A and B does the identity: $$ (A^{-1} + B^{-1})^{-1} = A - A(A+B)^{-1}A $$ hold? My supervisor suggested that they are equal but I haven't been able to prove this and in the ...
0
votes
0answers
43 views

How is $\sum_{i=0}^{\log n -1} 2^i(\log n -i) =\sum_{i=1}^{\log n} i\frac n {2^i} $

I found the following in a textbook: $$\sum_{i=0}^{\log n -1} 2^i(\log n -i) =\sum_{i=1}^{\log n} i\frac n {2^i} $$ It's a summation of a chart, the explanation for this was to "flip the chart", I ...
2
votes
0answers
70 views

system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ ...
0
votes
2answers
47 views

Divergence of laplacian

It seems to me that, in some derivations on fluid dynamic books I am reading, the identity $$\nabla \cdot (\nabla^2 u) = 0 $$ where $u$ is a vector field, is used. Does this identity exist? Is it ...
1
vote
0answers
25 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
0
votes
1answer
36 views

A function satisfies the identity … find another identity that $f(x)$ satisfies.

A function satisfies the identity $f(x) + 2f\left( \frac{1}{x} \right) = 2x+1$. By replacing all instances of $x$ with $\frac{1}{x}$, find another identity that $f(x)$ satisfies. I have ...
4
votes
0answers
46 views

Product of two binomial coefficients in terms of linear combinations of binomial coefficients [duplicate]

For $n,a,b$ natural numbers with $a+b \leq n$, can we find positive rational numbers $c_1, \ldots, c_{a+b}$ such that $$ {n \choose a}{n \choose b} = \sum_{k=1}^{a+b}c_k{n \choose k}? $$ Is there a ...
4
votes
3answers
116 views

Replacing in equation introduces more solutions

Let's say I have an equation $y=2-x^2-y^2$. now, since I know that $y$ is exactly the same as $2-x^2-y^2$ I can create the following, equation by replacing $y$ with $2-x^2-y^2$. ...
1
vote
0answers
52 views

Identity of two sums

I was trying to solve https://erdos.sdslabs.co/problems/31 The positive real numbers $x_0,x_1,x_2,\dotsc,x_m$ satisfy $x_0=x_m$ and $x_{i-1}+\frac{k}{x_{i-1}} = kx_i+\frac{1}{x_i}.$ Let ...
0
votes
0answers
64 views

Asymptotics relationship from algebric identity?

Background We start with the identity: $$ \sum_{r=1}^n r \ln r + \ln (r-1)! = n\ln n! $$ $$ \implies \sum_{r=1}^n \frac{r}{n} \ln r + \frac{1}{n} \ln\Gamma(r) = \ln n!$$ $$ \implies ...
0
votes
1answer
53 views

Is $f(x+a) - f(a) = f(x) + f'(a) x$ an identity?

Given a differentiable function of $x$, denoted by $f(x)$; is $f(x+a) - f(a) = f(x) + f'(a) x$ an identity? For example, if $f(x)=x^2$, then it gives $(x+a)^2 - a^2 = x^2 + 2ax$, which is true. So, ...
3
votes
1answer
36 views

Computerized search for integer identities. Suggestion of representation

I'm writing an app in Nim to search for curious integer identities such as those ones listed on:MathWorld: Divisor Function where it says "curious identities derived using modular theory". The ...
-1
votes
4answers
60 views

Proving algebraic identities

Could someone show how: $$\binom{n}{r}+\binom{n+1}{r}+\binom{n+2}{r}=\binom{n+3}{r+1}?$$ I tried expanding but in the end nothing really got cancelled to prove the identity.
0
votes
1answer
17 views

Trigonometric Identity using sin and csc

If $x$ is in the interval $[0, {\pi \over 2}]$ and $y$ is in the interval $[{\pi \over 2}, \pi]$ and $tan x={4 \over 3}$ and $csc y={13 \over 5}$, evaluate $$sin(x+y)$$ The final answer is supposed ...
3
votes
4answers
44 views

Identity of $8\sin^2(t)\cos^2(t)$

I know this probably has a simple answer, but I am having trouble understanding the steps to find the identity for this problem. This is the answer I was provided: $$8\sin^2(x)\cos^2(x) = ...
0
votes
2answers
50 views

Examples of 3d visual proofs

I am looking for examples of three dimensional constructible proofs. By this I mean activities such as steps in proving $1^2+2^2+\cdots+n^2=n(n+1)(2n+1)/6$. In this construction the identity is proven ...