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人気のある三角関数 >

証明する (cos(x)cot(x))/(1-sin(x))-1=csc(x)

解

証明する

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1 = csc (x)

解

真

解答ステップ

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1 = csc (x)

左側を操作する

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1

サイン, コサインで表わす

- 1 + \frac{cos ( x ) cot ( x )}{1 - sin ( x )}

基本的な三角関数の公式を使用する:

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

= - 1 + \frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

簡素化

- 1 + \frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )} : \frac{- sin ( x ) ( 1 - sin ( x ) ) + cos ^{2} ( x )}{sin ( x ) ( 1 - sin ( x ) )}

- 1 + \frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

\frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )} = \frac{cos ^{2} ( x )}{sin ( x ) ( 1 - sin ( x ) )}

\frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

乗じる

cos (x) \frac{cos ( x )}{sin ( x )} : \frac{cos ^{2} ( x )}{sin ( x )}

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

分数を乗じる:

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

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

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

cos (x) cos (x)

指数の規則を適用する:

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

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

= cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= cos^{2} (x)

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

= \frac{\frac{c o s ^{2} ( x )}{s i n ( x )}}{1 - sin ( x )}

分数の規則を適用する:

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

= \frac{cos ^{2} ( x )}{sin ( x ) ( 1 - sin ( x ) )}

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

元を分数に変換する:

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

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

分母が等しいので, 分数を組み合わせる:

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

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

乗算:

1 \cdot sin (x) = sin (x)

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

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

= \frac{cos ^{2} ( x ) - ( 1 - sin ( x ) ) sin ( x )}{( 1 - sin ( x ) ) sin ( x )}

三角関数の公式を使用して書き換える

\frac{cos ^{2} ( x ) - ( 1 - sin ( x ) ) sin ( x )}{( 1 - sin ( x ) ) sin ( x )}

ピタゴラスの公式を使用する:

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

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

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

簡素化

\frac{1 - sin ^{2} ( x ) - ( 1 - sin ( x ) ) sin ( x )}{( 1 - sin ( x ) ) sin ( x )} : \frac{1}{sin ( x )}

\frac{1 - sin ^{2} ( x ) - ( 1 - sin ( x ) ) sin ( x )}{( 1 - sin ( x ) ) sin ( x )}

拡張

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

1 - sin^{2} (x) - (1 - sin (x)) sin (x)

= 1 - sin^{2} (x) - sin (x) (1 - sin (x))

拡張

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

- sin (x) (1 - sin (x))

分配法則を適用する:

a (b - c) = ab - a c

a = - sin (x), b = 1, c = sin (x)

= - sin (x) \cdot 1 - (- sin (x)) sin (x)

マイナス・プラスの規則を適用する

- (- a) = a

= - 1 \cdot sin (x) + sin (x) sin (x)

簡素化

- 1 \cdot sin (x) + sin (x) sin (x) : - sin (x) + sin^{2} (x)

- 1 \cdot sin (x) + sin (x) sin (x)

1 \cdot sin (x) = sin (x)

1 \cdot sin (x)

乗算:

1 \cdot sin (x) = sin (x)

= sin (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

指数の規則を適用する:

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

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

数を足す:

1 + 1 = 2

= sin^{2} (x)

= - sin (x) + sin^{2} (x)

= - sin (x) + sin^{2} (x)

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

簡素化

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

1 - sin^{2} (x) - sin (x) + sin^{2} (x)

条件のようなグループ

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

類似した元を足す:

- sin^{2} (x) + sin^{2} (x) = 0

= - sin (x) + 1

= - sin (x) + 1

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

共通因数を約分する:

- sin (x) + 1

= \frac{1}{sin ( x )}

= \frac{1}{sin ( x )}

= \frac{1}{sin ( x )}

三角関数の公式を使用して書き換える

基本的な三角関数の公式を使用する:

sin (x) = \frac{1}{csc ( x )}

\frac{1}{\frac{1}{c s c ( x )}}

簡素化

\frac{1}{\frac{1}{c s c ( x )}}

分数の規則を適用する:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ( x )}{1}

規則を適用

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

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+tan(x))^2=sec^2(x)+2tan(x)

p ro v e (1 + tan (x))^{2} = sec^{2} (x) + 2 tan (x)

証明する (sec^2(x))/(2tan(x))=csc(2x)

p ro v e \frac{sec ^{2} ( x )}{2 tan ( x )} = csc (2 x)

証明する (sec^2(x)-1)csc(x)=tan(x)sec(x)

p ro v e (sec^{2} (x) - 1) csc (x) = tan (x) sec (x)

証明する cos(x-pi/2)=sin(x)

p ro v e cos (x - \frac{π}{2}) = sin (x)

証明する (1+tan^2(x))cot(x)=sec(x)csc(x)

p ro v e (1 + tan^{2} (x)) cot (x) = sec (x) csc (x)