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

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0
votes
3answers
24 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
39 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 ...
7
votes
2answers
56 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
37 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
583 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
41 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
69 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
35 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
22 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
35 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
104 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
50 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
62 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
33 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
58 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
16 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
42 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
43 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 ...
0
votes
1answer
38 views

Let $a$, $b$, and $c$ be positive real numbers with $a<b<c$ such that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=216$. Find $a+2b+3c$

Source: AoPS My attempt: $$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\implies144=50+2(ab+bc+ca) $$ $$\implies ab+bc+ca=47$$ and ...
1
vote
2answers
40 views

How can I find $\cos(\theta)$ with $\sin(\theta)$?

If $\sin^2x$ + $\sin^22x$ + $\sin^23x$ = 1, what does $\cos^2x$ + $\cos^22x$ + $\cos^23x$ equal? My attempted (and incorrect) solution: $\sin^2x$ + $\sin^22x$ + $\sin^23x$ = $\sin^26x$ = 1 ...
1
vote
2answers
35 views

Proofs of basilar powers identities

We all know that a simple and intuitive way to show what $2^n$ is (for $n$ an integer number) is to write it as $$2^n = \underbrace{2\times 2\times 2\times \cdots \times 2}_{n\ \text{times}}$$ My ...
3
votes
2answers
61 views

Combinatorics Identity about Catalan numbers.

I need to prove this identity: $\sum_{k=0}^n \frac{1}{k+1}{2k \choose k}{2n-2k \choose n-k}={2n+1 \choose n}$ without using the identity: $C_{n+1}=\sum_{k=0}^n C_kC_{n-k}$. Can't figure out how ...
1
vote
0answers
20 views

Is it possible to write identity: $(x(y^2-z^2)-y).(u(v^2-w^2)-v))=a(b^2-c^2)-b$?

Is it possible to write identity for $$ \{x(y^2-z^2)-y\}.\{u(v^2-w^2)-v)\}=a(b^2-c^2)-b $$ in integers similar to the identity $$ (x^2+y^2)(u^2+v^2)=(xu+yv)^2+(xv-yu)^2 $$ If possible, what can we ...
1
vote
0answers
33 views

Show that these two identities are equivalent

As an answer to the question Proof that for $a>0$ and $a + 1/a$ element of $\mathbb{Z}$, $a^n + 1/a^n$ is always element of $\mathbb{Z}$ by induction, user236182 gave this answer: ...
0
votes
0answers
37 views

Why is $(\sum\limits_{i=1}^n i)^2 =\sum\limits_{i=1}^n i^3 $ [duplicate]

I know that both evaluates to $\frac{(n^2)(n+1)^2}{4}$ but is it only a coincidence or is there some combinatorial reason?
0
votes
1answer
57 views

Give an example of sets $X,Y$ and functions $f:X→Y$ and $g:Y →X$ such that $g \circ f = id_X$

Give an example of sets $X,Y$ and functions $f:X→Y$ and $g:Y →X$ such that $g \circ f = id_X$ but $f$ is not invertible, and $id_X$ is the identity function of $x$. I am not sure how to even start ...
0
votes
1answer
12 views

Observation about convergent functions?

I noticed this first in the case of the function $\arctan x + \arctan \frac{1}{x} = \frac{\pi}{2}$. This identity is true for all $x$, and it got me thinking about functions that are asymptomatic in ...
3
votes
2answers
46 views

How show this $\overrightarrow {AC}\cdot\overrightarrow {BD}=\frac{1}{2}[(b^2+d^2)-(a^2+c^2)]$

In Convex quadrilateral $ABCD$,such $|AB|=a,|BC|=b,|CD|=c,|DA|=d$, show that $$\overrightarrow {AC}\cdot\overrightarrow {BD}=\dfrac{1}{2}[(b^2+d^2)-(a^2+c^2)]$$ I have one methods to solve this ...
0
votes
1answer
16 views

Simple matrix (vector) identity proof

During my experiments with matrices I found the following simple identity $$a^{T}Cabb^{T}=ba^{T}Cab^{T}$$ where $a(m,1),b(m,1)$ are the column vectors, and $C(m,m)$ is a regular matrix. How to prove ...
1
vote
1answer
40 views

Identity involving 2- and 3-cycles in group $S_3$

I would like to prove that a product of any 2-cycle $\sigma_2$ in symmetric group $S_3$ with any 3-cycle $\sigma_3$ in $S_3$ satisfies the identity $\sigma_2 \sigma_3 = \sigma_3^{-1} \sigma_2$. I ...
1
vote
1answer
66 views

Generalizations of the pentagonal number theorem

Euler's pentagonal number theorem (see also the original paper and review by Jordan Bell) states $$ \prod_{n=1}^\infty (1 - q^n) = \sum_{k=-\infty}^{\infty} (-1)^{k} q^{(3k^2 - k)/2}, $$ where $k \, ...
0
votes
0answers
20 views

Solve equation from a summation of tan(arcsin)

Hello I am having trouble solving this equation. To solve for p $$\ X(p) = 2p \sum_{k=1}^{n} \frac{z_{k}v_{k}}{\sqrt{[1-p^2v^2_k]}} $$ I know I can use a trig identity to simplify to $$\ X(p) = 2 ...
0
votes
2answers
52 views

Proving $a^n-b^n=(a-b)\sum_{k=1}^{n}a^{n-k} b^{k-1}$ with induction [duplicate]

How can I prove this using induction. I proved for n=1 but now I'm feeling confused while I'm trying to prove for n+1 because of how the summation develops $$ a^n-b^n=(a-b)\sum_{k=1}^{n}a^{n-k} ...
1
vote
2answers
44 views

Need help in solving log identity

I have the quantity $\frac{\log(x)}{\log(x) + \log(y)}$ What I need is to calculate $\log(\frac{x}{x+y})$ Is it possible? Any further help would be much appreciated!
3
votes
1answer
56 views

Why does rearranging Euler's identity in this manner result in a false statement?

I placed each of the following steps into Wolfram alpha after working it out in my head. All steps prior to the one marked with the (*) held true. $e^{i \pi} = -1$ $e^{2i \pi} = 1$ $\ln(e^{2i \pi}) ...
2
votes
1answer
43 views

A high-level reason that $(a \times b) \cdot ((b \times c) \times (c \times a)) = (a \cdot (b \times c))^2$

I can do the algebra to prove this identity: $$(\mathbf{a} \times \mathbf{b}) \cdot ((\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a})) = (\mathbf{a} \cdot (\mathbf{b} \times ...
0
votes
1answer
53 views

Is there a way to turn an alternating series/sum into a strictly positive one?

Let's say I have an alternating sum $$S = \sum_i (-1)^i a_i,$$ where the coefficients $a_i$ are all positive rational numbers, and we assume $S > 0$. What methods are available to obtain a derived ...
2
votes
2answers
45 views

Is there a simple way to prove this identity using double summation?

I am trying to prove that $$\sum_{k=0}^n\sum_{r=0}^k{n \choose k}{k \choose r}(-1)^{n+k}A_r=\sum_{r=0}^n\sum_{k=r}^n{n \choose k}{k \choose r}(-1)^{n+k}A_r$$ By writing out the full sum I can see ...
3
votes
0answers
49 views

Show that $\text{deg}(M_n(\mathbb{K})) = n$, where $\mathbb{K}$ is a field.

Definition: Let $A$ be a ring and $Z=Z(A)$ its center. We say that $t \in A$ is algebraic over $Z$ if there exist $z_0,z_1, \ldots , z_n \in Z$ such that $$z_0+z_1t+ \cdots + z_n t^n = 0 \quad ...
7
votes
2answers
168 views

Proving a Polynomial Identity

Prove that $$\sum_{i=1}^{n} \dfrac{{r_{i}}^k P(x)}{P'(r_{i})(x-r_{i})} = x^k$$ where $P(x)$ is an $n$ degree polynomial having distinct roots $\{ r_{i} \}_{i=1}^{n}$ and $k$ is an ...
0
votes
1answer
64 views

Trig identities: write answer in terms of sine and cosine

I attached an image but just in case it doesn't show up properly, the prompt is to write $$\frac{\csc(x)\cot(x)}{\sec(x)}$$ in terms of sine and cosine. What I don't understand is that the prompt is ...
4
votes
3answers
174 views

Algebraic and combinatorial proof of an identity

For any two integers $2 \le k \le n-2$, there is the identity $$\dbinom{n}{2} = \dbinom{k}{2} + k(n-k) + \dbinom{n-k}{2}.$$ a) Give an algebraic proof of this identity, writing the binomial ...
1
vote
1answer
67 views

If $[A,A]A[\lambda,A] = 0$ then $\lambda \in Z(A).$

Suppose that $A$ is a unital ring and $([A,A]) = A.$ If $[A,A]A[\lambda,A] = 0$ prove that $\lambda \in Z(A).$ Comments: This is part of an exercise I'm doing, I'm posting this part because I am ...
1
vote
4answers
76 views

Trignometric Identities and Equations

For the following problem(s) I cannot get any answer(s). I would appreciate your help very much. $$\tan { \theta -\sec { \theta } =\sqrt { 3 } } $$ TI get 30 degrees as the reference angle. What ...
0
votes
0answers
36 views

Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y \in A$.

Let $A = M_2(C)$, where $C$ is any commutative ring. Define $E: A \longrightarrow A$ by $E(x) = x - tr(x)Id_2$, where $tr(x)$ denotes the trace of $x$. Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y ...
1
vote
1answer
91 views

Proof of identity A = A or 1 = 1

Is 1 = 1 an assumption? I feel it's a very good assumption, but is there a proof for it? Imagine a world where people were contesting it, where equivalence wasn't a common sense concept. In reality no ...
3
votes
1answer
82 views

an interesting q-series and a certain continued-fraction

My aim is to find a rigorous proof of the following conjectured identity.Given ...
2
votes
2answers
37 views

Using identities from Euclid's algorithm to solve problems

I have been given the following problems (a) Use the $GCD$ algorithm to compute the greatest common divisor of $546$ and $416$. (b) Use your working for the previous part to express ...