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人気のある三角関数 >

証明する ((csc^2(x)))/(cot(x))=csc(x)sec(x)

解

証明する

\frac{( csc ^{2} ( x ) )}{cot ( x )} = csc (x) sec (x)

解

真

解答ステップ

\frac{csc ^{2} ( x )}{cot ( x )} = csc (x) sec (x)

左側を操作する

\frac{csc ^{2} ( x )}{cot ( x )}

サイン, コサインで表わす

\frac{csc ^{2} ( x )}{cot ( x )}

基本的な三角関数の公式を使用する:

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

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{cot ( x )}

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

分数の規則を適用する:

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

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

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

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

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

乗じる

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

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

分数を乗じる:

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

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

乗算:

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

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

共通因数を約分する:

sin (x)

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

= \frac{\frac{1}{s i n ( x )}}{cos ( x )}

分数の規則を適用する:

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

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

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

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

基本的な三角関数の公式を使用する:

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

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

簡素化

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

乗じる

\frac{1}{sec ( x )} \cdot \frac{1}{csc ( x )} : \frac{1}{sec ( x ) csc ( x )}

\frac{1}{sec ( x )} \cdot \frac{1}{csc ( x )}

分数を乗じる:

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

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

数を乗じる:

1 \cdot 1 = 1

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

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

分数の規則を適用する:

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

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

規則を適用

\frac{a}{1} = a

= sec (x) csc (x)

sec (x) csc (x)

sec (x) csc (x)

= csc (x) sec (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 csc^2(x)(tan^2(x))-1=tan^2(x)

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

証明する sin(x)cos(x)+sin(3x)sec(x)=tan(x)

p ro v e sin (x) cos (x) + sin (3 x) sec (x) = tan (x)

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

p ro v e (1 + sec (x)) (1 - cos (x)) = sin (x) tan (x)

証明する sec(x)*sin(x)=tan(x)

p ro v e sec (x) \cdot sin (x) = tan (x)

証明する tan(x)csc(x)sec(x)=tan^2(x)+1

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