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人気のある三角関数 >

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

解

証明する

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

解

真

解答ステップ

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

右側を操作する

cos^{2} (a) + (cot (a) cos (a))^{2}

サイン, コサインで表わす

(cos (a) cot (a))^{2} + cos^{2} (a)

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

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

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

乗じる

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

cos (a) \frac{cos ( a )}{sin ( a )}

分数を乗じる:

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

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

cos (a) cos (a) = cos^{2} (a)

cos (a) cos (a)

指数の規則を適用する:

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

cos (a) cos (a) = cos^{1 + 1} (a)

= cos^{1 + 1} (a)

数を足す:

1 + 1 = 2

= cos^{2} (a)

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

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

指数の規則を適用する:

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

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

(cos^{2} (a))^{2} : cos^{4} (a)

指数の規則を適用する:

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

= cos^{2 \cdot 2} (a)

数を乗じる:

2 \cdot 2 = 4

= cos^{4} (a)

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

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

元を分数に変換する:

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

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

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

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

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

因数

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

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

因数

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

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

指数の規則を適用する:

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

cos^{4} (a) = cos^{2} (a) cos^{2} (a)

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

共通項をくくり出す

cos^{2} (a)

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

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

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

三角関数の公式を使用して書き換える

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

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} ( a )}{sin ^{2} ( a )}

簡素化

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

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

三角関数の公式を使用して書き換える

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

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 ( a ) cot ( a )}{sin ( a )}

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

基本的な三角関数の公式を使用する:

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

cot (a) cot (a)

簡素化

cot (a) cot (a) : cot^{2} (a)

cot (a) cot (a)

指数の規則を適用する:

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

cot (a) cot (a) = cot^{1 + 1} (a)

= cot^{1 + 1} (a)

数を足す:

1 + 1 = 2

= cot^{2} (a)

cot^{2} (a)

cot^{2} (a)

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 (1-sin^2(t))(1+tan^2(t))=1

p ro v e (1 - sin^{2} (t)) (1 + tan^{2} (t)) = 1

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p ro v e 1 - cosh^{2} (x) = - sinh^{2} (x)

証明する cot^2(θ)+1=csc^2(θ)

p ro v e cot^{2} (θ) + 1 = csc^{2} (θ)

証明する 2cos(θ)tan(θ)csc(θ)=2

p ro v e 2 cos (θ) tan (θ) csc (θ) = 2

証明する ((cos^2(x)+4cos(x)+4))/((cos(x)+2))=((2sec(x)+1))/(sec(x))

p ro v e \frac{( cos ^{2} ( x ) + 4 cos ( x ) + 4 )}{( cos ( x ) + 2 )} = \frac{( 2 sec ( x ) + 1 )}{sec ( x )}