# Tagged Questions

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### $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l}$

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l}$ for every $m, n, l >0$.
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### Completing sets of numbers solely with trigonometric functions and an initial zero?

Last week an extra-curricular math academy I attend gave us this question as a challenge: You start with $0$, and the only functions you can do are $\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}$ ...
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### An upper bound on certain finite trigonometric series given a lower bound

Let $f$ be the function $f(x)=1+a\sin{x}+b\cos x+c\sin{(2x)}+d\cos{(2x)}$, where $a,b,c,d$ are arbitrary real numbers. Prove that if $f(x)>0$ for all $x\in \mathbb R$, then $f(x)<3$ for all ...
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### How find this $\sum_{i=0}^{5}\frac{1}{2+\cos{\left(x+\frac{i\pi}{3}\right)}}\cdot \frac{1}{2+\cos{\left(x+\frac{(i+1)\pi}{3}\right)}}$

Find this follow function $f(x)$ range ,where $x\in R$, $$f(x)=\sum_{i=0}^{5}\dfrac{1}{2+\cos{\left(x+\dfrac{i\pi}{3}\right)}}\cdot \dfrac{1}{2+\cos{\left(x+\dfrac{(i+1)\pi}{3}\right)}}$$ or ...
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### Competition Math Geometry Problem

Note: I am paraphrasing this problem Consider a quadrilateral with 3 sides of equal length, and one longer side. This quadrilateral also has equal diagonals, both of which are equal in length to the ...
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### Convex Quadrilateral: $\dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D$

Problem Let $ABCD$ be a convex quadrilateral with no right angles. Show that $$\dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D.$$ ...
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### How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
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### Macedonia National Olympiad 2010

Problem The point O is the center of the circumscribed circle of the acute-angled triangle ABC. The line AO cuts the side BC in point N, and the line BO cuts the side AC at point M. Prove that if ...
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### Prove this trigonometric identity in quadrilateral

If $\alpha,\beta,\gamma,\delta$ are angles in quadrilateral different from $90^\circ$, prove the following:  ...
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### Another Trigonometry problem, sum of products of sine function over partitions of N

I don't know how to write the summation symbol so I'm providing you the original link to problem http://www.codechef.com/OCT11/problems/PARSIN .My approach to solve this problem is first reduce the ...
### Average and minimum Values of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$, $\forall x \in \mathbb{R}$
A problem was asked at Putnam Competition in 2003 (Problem 3), about finding the minimum Value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ where $x$ is Real. the question paper and ...
Problem 63 of the 2001 St. Petersburg Mathematical Olympiad, Second Round, 11th grade: Are there three different numbers $x, y, z$ in $[0,\pi/2]$ such that the numbers $\sin x$, $\sin y$, $\sin z$, ...