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人気のある三角関数 >

証明する sec^2(θ)cot^2(θ)-1=cot^2(θ)

解

証明する

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

解

真

解答ステップ

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

左側を操作する

sec^{2} (θ) cot^{2} (θ) - 1

サイン, コサインで表わす

- 1 + cot^{2} (θ) sec^{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{cos ( θ )}{sin ( θ )})^{2} sec^{2} (θ)

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

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

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

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

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

(\frac{1}{cos ( θ )})^{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} ( θ )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

分数を乗じる:

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

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

共通因数を約分する:

cos^{2} (θ)

= \frac{1}{sin ^{2} ( θ )}

= - 1 + \frac{1}{sin ^{2} ( θ )}

元を分数に変換する:

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

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

乗算:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

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

右側を操作する

cot^{2} (θ)

サイン, コサインで表わす

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 )}

= (\frac{cos ( θ )}{sin ( θ )})^{2}

簡素化

(\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{cos ^{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 )}

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} ( θ )}{sin ^{2} ( θ )}

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

両辺を同じ形式にできることを証明した

\Rightarrow 真

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