証明する cot^2(θ)(1-cos^2(θ))=cos^2(θ)

解

証明する

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

真

解答ステップ

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

左側を操作する

cot^{2} (θ) (1 - cos^{2} (θ))

サイン, コサインで表わす

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

簡素化

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

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

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

共通因数を約分する:

sin^{2} (θ)

= cos^{2} (θ)

= cos^{2} (θ)

両辺を同じ形式にできることを証明した

\Rightarrow 真