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人気のある三角関数 >

証明する cot^2(θ)(1+tan^2(θ))=csc^2(θ)

解

証明する

cot^{2} (θ) (1 + tan^{2} (θ)) = csc^{2} (θ)

解

真

解答ステップ

cot^{2} (θ) (1 + tan^{2} (θ)) = csc^{2} (θ)

左側を操作する

cot^{2} (θ) (1 + tan^{2} (θ))

サイン, コサインで表わす

(1 + tan^{2} (θ)) cot^{2} (θ)

基本的な三角関数の公式を使用する:

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

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

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{sin ( θ )}{cos ( θ )})^{2}) (\frac{cos ( θ )}{sin ( θ )})^{2}

簡素化

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

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

指数の規則を適用する:

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

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

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

(\frac{cos ( θ )}{sin ( θ )})^{2}

指数の規則を適用する:

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

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

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

分数を乗じる:

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

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

結合

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

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

元を分数に変換する:

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

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

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 cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ )}

乗算:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

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

= \frac{\frac{c o s ^{2} ( θ ) + s i n ^{2} ( θ )}{c o s ^{2} ( θ )} cos ^{2} ( θ )}{sin ^{2} ( θ )}

乗じる

cos^{2} (θ) \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ )} : cos^{2} (θ) + sin^{2} (θ)

cos^{2} (θ) \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ )}

分数を乗じる:

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

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

共通因数を約分する:

cos^{2} (θ)

= cos^{2} (θ) + sin^{2} (θ)

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

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

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

三角関数の公式を使用して書き換える

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ^{2} ( θ )}

ピタゴラスの公式を使用する:

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

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

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 ( θ )} ) ^{2}}

簡素化

\frac{1}{( \frac{1}{c s c ( θ )} ) ^{2}}

(\frac{1}{csc ( θ )})^{2} = \frac{1}{csc ^{2} ( θ )}

(\frac{1}{csc ( θ )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{csc ^{2} ( θ )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( θ )}

= \frac{1}{\frac{1}{c s c ^{2} ( θ )}}

分数の規則を適用する:

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

= \frac{csc ^{2} ( θ )}{1}

規則を適用

\frac{a}{1} = a

= csc^{2} (θ)

csc^{2} (θ)

csc^{2} (θ)

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

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