ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある三角関数 >

証明する 1/(1-sin(r))=sec^2(r)+sec(r)tan(r)

解

証明する

\frac{1}{1 - sin ( r )} = sec^{2} (r) + sec (r) tan (r)

解

真

解答ステップ

\frac{1}{1 - sin ( r )} = sec^{2} (r) + sec (r) tan (r)

右側を操作する

sec^{2} (r) + sec (r) tan (r)

サイン, コサインで表わす

sec^{2} (r) + sec (r) tan (r)

基本的な三角関数の公式を使用する:

sec (x) = \frac{1}{cos ( x )}

= (\frac{1}{cos ( r )})^{2} + \frac{1}{cos ( r )} tan (r)

基本的な三角関数の公式を使用する:

tan (x) = \frac{sin ( x )}{cos ( x )}

= (\frac{1}{cos ( r )})^{2} + \frac{1}{cos ( r )} \cdot \frac{sin ( r )}{cos ( r )}

簡素化

(\frac{1}{cos ( r )})^{2} + \frac{1}{cos ( r )} \cdot \frac{sin ( r )}{cos ( r )} : \frac{1 + sin ( r )}{cos ^{2} ( r )}

(\frac{1}{cos ( r )})^{2} + \frac{1}{cos ( r )} \cdot \frac{sin ( r )}{cos ( r )}

(\frac{1}{cos ( r )})^{2} = \frac{1}{cos ^{2} ( r )}

(\frac{1}{cos ( r )})^{2}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cos ^{2} ( r )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( r )}

\frac{1}{cos ( r )} \cdot \frac{sin ( r )}{cos ( r )} = \frac{sin ( r )}{cos ^{2} ( r )}

\frac{1}{cos ( r )} \cdot \frac{sin ( r )}{cos ( r )}

分数を乗じる:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot sin ( r )}{cos ( r ) cos ( r )}

乗算:

1 \cdot sin (r) = sin (r)

= \frac{sin ( r )}{cos ( r ) cos ( r )}

cos (r) cos (r) = cos^{2} (r)

cos (r) cos (r)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

cos (r) cos (r) = cos^{1 + 1} (r)

= cos^{1 + 1} (r)

数を足す:

1 + 1 = 2

= cos^{2} (r)

= \frac{sin ( r )}{cos ^{2} ( r )}

= \frac{1}{cos ^{2} ( r )} + \frac{sin ( r )}{cos ^{2} ( r )}

規則を適用

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + sin ( r )}{cos ^{2} ( r )}

= \frac{1 + sin ( r )}{cos ^{2} ( r )}

= \frac{1 + sin ( r )}{cos ^{2} ( r )}

三角関数の公式を使用して書き換える

\frac{1 + sin ( r )}{cos ^{2} ( r )}

ピタゴラスの公式を使用する:

cos^{2} (r) + sin^{2} (r) = 1

cos^{2} (r) = 1 - sin^{2} (r)

= \frac{1 + sin ( r )}{1 - sin ^{2} ( r )}

因数

1 - sin^{2} (r) : (1 + sin (r)) (1 - sin (r))

1 - sin^{2} (r)

2乗の差の公式を適用する:

r^{2} - y^{2} = (r + y) (r - y)

1 - sin^{2} (r) = (1 + sin (r)) (1 - sin (r))

= (1 + sin (r)) (1 - sin (r))

= \frac{1 + sin ( r )}{( 1 + sin ( r ) ) ( 1 - sin ( r ) )}

共通因数を約分する:

1 + sin (r)

= \frac{1}{1 - sin ( r )}

= \frac{1}{1 - sin ( r )}

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

証明する 1/(sec(x))=sec(x)-tan(x)sin(x)

p ro v e \frac{1}{sec ( x )} = sec (x) - tan (x) sin (x)

証明する sin(a+b)sin(a-b)=sin^2(b)-sin^2(a)

p ro v e sin (a + b) sin (a - b) = sin^{2} (b) - sin^{2} (a)

証明する sec(b)+tan(b)=(cos(b))/(1-sin(b))

p ro v e sec (b) + tan (b) = \frac{cos ( b )}{1 - sin ( b )}

証明する cot^4(x)+cot^2(x)=csc^4(x)-csc^2(x)

p ro v e cot^{4} (x) + cot^{2} (x) = csc^{4} (x) - csc^{2} (x)

証明する 1-tanh^2(x)=sech^2(x)

p ro v e 1 - tanh^{2} (x) = sech^{2} (x)