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人気のある三角関数 >

証明する cot(x)-cot(x)cos^2(x)=sin(x)cos(x)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

cot (x) - cot (x) cos^{2} (x)

サイン, コサインで表わす

cot (x) - cos^{2} (x) cot (x)

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 ( x )}{sin ( x )} - cos^{2} (x) \frac{cos ( x )}{sin ( x )}

簡素化

\frac{cos ( x )}{sin ( x )} - cos^{2} (x) \frac{cos ( x )}{sin ( x )} : \frac{cos ( x ) - cos ^{3} ( x )}{sin ( x )}

\frac{cos ( x )}{sin ( x )} - cos^{2} (x) \frac{cos ( x )}{sin ( x )}

cos^{2} (x) \frac{cos ( x )}{sin ( x )} = \frac{cos ^{3} ( x )}{sin ( x )}

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

分数を乗じる:

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

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

cos (x) cos^{2} (x) = cos^{3} (x)

cos (x) cos^{2} (x)

指数の規則を適用する:

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

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

= cos^{1 + 2} (x)

数を足す:

1 + 2 = 3

= cos^{3} (x)

= \frac{cos ^{3} ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )} - \frac{cos ^{3} ( x )}{sin ( x )}

規則を適用

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

= \frac{cos ( x ) - cos ^{3} ( x )}{sin ( x )}

= \frac{cos ( x ) - cos ^{3} ( x )}{sin ( x )}

= \frac{cos ( x ) - cos ^{3} ( x )}{sin ( x )}

因数

\frac{cos ( x ) - cos ^{3} ( x )}{sin ( x )} : \frac{cos ( x ) ( 1 - cos ^{2} ( x ) )}{sin ( x )}

\frac{cos ( x ) - cos ^{3} ( x )}{sin ( x )}

因数

cos (x) - cos^{3} (x) : cos (x) (1 - cos^{2} (x))

cos (x) - cos^{3} (x)

指数の規則を適用する:

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

cos^{3} (x) = cos (x) cos^{2} (x)

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

共通項をくくり出す

cos (x)

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

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

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

三角関数の公式を使用して書き換える

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

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} ( x ) cos ( x )}{sin ( x )}

共通因数を約分する:

sin (x)

= sin (x) cos (x)

= sin (x) cos (x)

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 \frac{1}{sec ( θ )} = arcsec (θ)

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p ro v e \frac{( cot ( x ) )}{( csc ( x ) )} = cos (x)

証明する ((sec^2(t)))/(sec^2(t)-1)=csc^2(t)

p ro v e \frac{( sec ^{2} ( t ) )}{sec ^{2} ( t ) - 1} = csc^{2} (t)

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p ro v e sin^{2} (x) = cos (2 x) + 2

証明する 1+cos^2(x)=2-sin^2(x)

p ro v e 1 + cos^{2} (x) = 2 - sin^{2} (x)