Discrete formulas such as the Faulhaber summations can be verified by evaluating them for a finite number of values. For example $$\sum_{k=1}^nk=\frac{n(n+1)}2$$ is validated by evaluating for $n=0,1,2$ (indeed $0=0,0+1=2/2,0+1+2=2\cdot3/2$) because both sides are quadratic polynomials. Can we ...

I have the following identity to verify: $$\sin(x+1)\sin(x+1) - \sin(x+2)\sin x = \sin^2(1).$$ I'm becoming more familiar with sum and difference formulas to some degree, but this one has stumped me. I don't know if I'm doing it right, even, but I have this so far: $$(\sin x \cos(1) + \cos x \...

I know both terms are the same because their graphs are, so I want to proof this equality using trigonometric identities. $$ \sin( A - B) = \sin A\cos B - \cos A\sin B $$ $$ \sin( A + B) = \sin A\cos B + \cos A\sin B $$ This is what I have got so far. I am not sure how I can get rid of the "-"...

Show that if $x$, $y$, and $z$ are consecutive terms of an arithmetic sequence, with $x \leq y \leq z$, and both sides of the equation are defined, then $$\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y.$$ I have no idea how to even start this problem, I'm stuck. Solution...

I have been trying to solve this question from a past exam for a very long time but don't know how to go about it. I tried making the LHS of the equation equal the identity but that didn't go far. Any help is appreciated. Let $(G, ∗, I)$ be a group. Let $x, y ∈ G$ both have finite order. Prov...

Solve for $A$ if $\sin 2x-\cos2x= \sqrt{2}\sin(2x+A\pi)$. I tried it with trigonometric identities but I can't solve it.

Can someone please help me understand this problem? I do not know what it means to find the identity so I don't know which steps to take first. (a-b)(a-1)(1-ab) = \begin{vmatrix} 1 & a & b & 1 \\ 1 & a & a & a \\a & 1 & ab & b \\ a & a & ab & 1 \end{vmatrix}.

Take this mathematical structure for example: (Z, #) where Z is the set of all integers and where a#b returns the larger of the two elements. In order to prove this is a group I must prove these four properties: -Closure -Associative -Identity -Inverse I am stuck on the Identity Element. As f...

Find all sets $S$ with at least one element such that for any two reals $a$ and $b$ with $a+b\in S$ we have $ab\in S$. Progress: If $ab\in S$ then $(b-1)a^2,2(b-2)a^2,\cdots$ is also present in the set as we can write $ab=a+(b-1)a,2a+(b-2)a,\cdots{}$.