# Quod Erat Demonstrandum

## 2020/02/28

### 正弦積

$\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o$

$\tan \theta \tan (90^o-\theta) \equiv 1$

$\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o$
$=(\tan 1^o\tan 89^o)(\tan 2^o\tan 88^o)\dots (\tan 44^o\tan 46^o)\tan 45^o$
$=1\times 1\times \dots \times 1$
$=1$

$\sin 1^o\sin 2^o\sin 3^o\dots \sin 88^o\sin 89^o$

(more…)

## 2019/05/10

### 線長乘積

Filed under: Pure Mathematics — johnmayhk @ 11:52 下午
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## 2019/02/01

### 帕斯卡三角某結果

Filed under: NSS,Pure Mathematics — johnmayhk @ 5:35 下午
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$f^{[1]}(x)=f(x)$

$f^{[2]}(x)=f(f(x))$ (more…)

## 2017/12/25

### 等邊三角形

Filed under: Junior Form Mathematics,Pure Mathematics — johnmayhk @ 12:13 下午
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e.g. 1

## 2017/06/23

### 實數問題複數解決

Filed under: mathematics,NSS — johnmayhk @ 3:43 下午
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$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$

## 2015/09/13

### 一式過

Filed under: Pure Mathematics — johnmayhk @ 5:19 下午
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$-1,1,-1,1,\dots$

$(-1)^n$

## 2015/08/27

### 某 monic 多項式

Filed under: NSS,Pure Mathematics — johnmayhk @ 10:09 上午
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## 2014/02/20

### 平方和

Filed under: Fun — johnmayhk @ 11:44 下午
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$a_1,a_2,a_3,\dots a_n$

$b=\sqrt{a_2^2+a_3^2+\dots +a_n^2}$

$(a_1+bi)^m=P+Qi$

## 2012/05/01

### 無聊談通項

$2,1,4,\frac{1}{2},8,\frac{1}{4},\dots$

## 2012/02/26

### 答網友：y=z^2

Filed under: HKALE,Pure Mathematics — johnmayhk @ 6:29 下午
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By considering

$(1+z)^8+(1-z)^8=0$

show that

$\displaystyle\sum_{k=0}^7 \tan^2\frac{(2k+1)\pi}{16}=56$

## 2012/01/20

### arctan,pi,complex numbers

Filed under: mathematics,Pure Mathematics — johnmayhk @ 12:02 下午
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$\pi=\tan^{-1}1+\tan^{-1}2+\tan^{-1}3$

$\tan^{-1}(\frac{1}{2})+\tan^{-1}(\frac{1}{3})=\frac{\pi}{4}$

（易知 $\tan^{-1}(\frac{1}{n})+\tan^{-1}(n)=\frac{\pi}{2}$，故上述兩式等價。）

## 2011/11/26

### 利用複數尋寶

Filed under: Fun,Pure Mathematics — johnmayhk @ 4:51 下午
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## 2011/11/18

### polar form

Filed under: Pure Mathematics — johnmayhk @ 11:46 上午
Tags: ,

Convert $z=-\cos \theta -i\sin\theta$ into polar form.

## 2011/03/02

### 當 x 接近零，(1+1/x)^x 如何？

Filed under: Pure Mathematics — johnmayhk @ 5:07 下午
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$\displaystyle \lim_{x\rightarrow \infty}(1+\frac{1}{x})^{x} = e$

$\displaystyle \lim_{x\rightarrow \infty}(2+\frac{1}{x})^{x} = ?$

$\displaystyle \lim_{x\rightarrow 0}(1+\frac{1}{x})^{x} = ?$

## 2009/10/26

### 利用圖像尋找非實根

Filed under: HKALE,NSS,Pure Mathematics — johnmayhk @ 8:24 下午
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