# Tagged Questions

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

7answers
94 views

### Why is this true $\frac{1-y^n}{1-y}=(1+y+y^2+…+ y^{n-1})$? [on hold]

I have a heard time seeing why is this true $\frac{1-y^n}{1-y}=(1+y+y^2+...+ y^{n-1})$ Could you show me some kind of proof, or an identity that would me to find this?
0answers
42 views

### Under what circumstances exactly is the identity element of a certain function unique?

I have a particular case in mind. Suppose there is some function $h$ which eliminates elements of a set so $h(x)\subseteq x \forall x\in P(\mathbb{N})$ Now imagine the (possibly) abelian group ...
3answers
412 views

### Combinatorial formulas and interpretations

I found that $$\sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! =s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} = \frac{(n+1)!}{n+1-s}$$ I proved this formula with induction, but I was wondering if there is a (...
2answers
77 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 ...
1answer
85 views

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 ...
1answer
31 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 (...
1answer
58 views

4answers
135 views

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

I need to establish the identity but am not sure how: $$\cos(\pi - \theta) = - \cos(\theta).$$
1answer
56 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 ...
2answers
43 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 theorem,...
4answers
57 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 ...
1answer
48 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 ...
1answer
32 views

1answer
39 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 ...
0answers
52 views