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人気のある三角関数 >

証明する cot^2(α)=cos^2(α)+(cot(α)cos(α))^2

解

証明する

cot^{2} (α) = cos^{2} (α) + (cot (α) cos (α))^{2}

解

真

解答ステップ

cot^{2} (α) = cos^{2} (α) + (cot (α) cos (α))^{2}

右側を操作する

cos^{2} (α) + (cot (α) cos (α))^{2}

サイン, コサインで表わす

(cos (α) cot (α))^{2} + cos^{2} (α)

基本的な三角関数の公式を使用する:

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

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

簡素化

(cos (α) \frac{cos ( α )}{sin ( α )})^{2} + cos^{2} (α) : \frac{cos ^{4} ( α ) + cos ^{2} ( α ) sin ^{2} ( α )}{sin ^{2} ( α )}

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

(cos (α) \frac{cos ( α )}{sin ( α )})^{2} = \frac{cos ^{4} ( α )}{sin ^{2} ( α )}

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

乗じる

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

cos (α) \frac{cos ( α )}{sin ( α )}

分数を乗じる:

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

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

cos (α) cos (α) = cos^{2} (α)

cos (α) cos (α)

指数の規則を適用する:

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

cos (α) cos (α) = cos^{1 + 1} (α)

= cos^{1 + 1} (α)

数を足す:

1 + 1 = 2

= cos^{2} (α)

= \frac{cos ^{2} ( α )}{sin ( α )}

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

指数の規則を適用する:

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

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

(cos^{2} (α))^{2} : cos^{4} (α)

指数の規則を適用する:

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

= cos^{2 \cdot 2} (α)

数を乗じる:

2 \cdot 2 = 4

= cos^{4} (α)

= \frac{cos ^{4} ( α )}{sin ^{2} ( α )}

= \frac{cos ^{4} ( α )}{sin ^{2} ( α )} + cos^{2} (α)

元を分数に変換する:

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

= \frac{cos ^{4} ( α )}{sin ^{2} ( α )} + \frac{cos ^{2} ( α ) sin ^{2} ( α )}{sin ^{2} ( α )}

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{cos ^{4} ( α ) + cos ^{2} ( α ) sin ^{2} ( α )}{sin ^{2} ( α )}

= \frac{cos ^{4} ( α ) + cos ^{2} ( α ) sin ^{2} ( α )}{sin ^{2} ( α )}

= \frac{cos ^{4} ( α ) + cos ^{2} ( α ) sin ^{2} ( α )}{sin ^{2} ( α )}

因数

\frac{cos ^{4} ( α ) + cos ^{2} ( α ) sin ^{2} ( α )}{sin ^{2} ( α )} : \frac{cos ^{2} ( α ) ( cos ^{2} ( α ) + sin ^{2} ( α ) )}{sin ^{2} ( α )}

\frac{cos ^{4} ( α ) + cos ^{2} ( α ) sin ^{2} ( α )}{sin ^{2} ( α )}

因数

cos^{4} (α) + cos^{2} (α) sin^{2} (α) : cos^{2} (α) (cos^{2} (α) + sin^{2} (α))

cos^{4} (α) + cos^{2} (α) sin^{2} (α)

指数の規則を適用する:

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

cos^{4} (α) = cos^{2} (α) cos^{2} (α)

= cos^{2} (α) cos^{2} (α) + cos^{2} (α) sin^{2} (α)

共通項をくくり出す

cos^{2} (α)

= cos^{2} (α) (cos^{2} (α) + sin^{2} (α))

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

簡素化

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

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

三角関数の公式を使用して書き換える

= \frac{cos ( α )}{sin ( α )} \cdot \frac{cos ( α )}{sin ( α )}

基本的な三角関数の公式を使用する:

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

\frac{cos ( α ) cot ( α )}{sin ( α )}

= cot (α) \frac{cos ( α )}{sin ( α )}

基本的な三角関数の公式を使用する:

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

cot (α) cot (α)

簡素化

cot (α) cot (α) : cot^{2} (α)

cot (α) cot (α)

指数の規則を適用する:

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

cot (α) cot (α) = cot^{1 + 1} (α)

= cot^{1 + 1} (α)

数を足す:

1 + 1 = 2

= cot^{2} (α)

cot^{2} (α)

cot^{2} (α)

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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p ro v e (1 - sin^{2} (x)) \cdot csc^{2} (x) = cot^{2} (x)

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p ro v e cot^{2} (θ) - cos^{2} (θ) = cot^{2} (θ) cos^{2} (θ)

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

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