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人気のある三角関数 >

証明する (tan(x)-cot(x))/(tan(x)+cot(x))=1-2cos^2(x)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

\frac{tan ( x ) - cot ( x )}{tan ( x ) + cot ( x )}

サイン, コサインで表わす

\frac{- cot ( x ) + tan ( x )}{cot ( x ) + tan ( 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{- \frac{c o s ( x )}{s i n ( x )} + tan ( x )}{\frac{c o s ( x )}{s i n ( x )} + tan ( x )}

基本的な三角関数の公式を使用する:

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

= \frac{- \frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}

簡素化

\frac{- \frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}} : \frac{- cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x ) + sin ^{2} ( x )}

\frac{- \frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}

結合

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

\frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )}

以下の最小公倍数:

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

sin (x)

または以下のいずれかに現れる因数で構成された式を計算する:

cos (x)

= sin (x) cos (x)

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

sin (x) cos (x)

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

の場合

:

分母と分子に以下を乗じる:

cos (x)

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

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

の場合

:

分母と分子に以下を乗じる:

sin (x)

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

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

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

= \frac{- \frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{s i n ( x ) c o s ( x )}}

結合

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

- \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )}

以下の最小公倍数:

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

sin (x)

または以下のいずれかに現れる因数で構成された式を計算する:

cos (x)

= sin (x) cos (x)

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

sin (x) cos (x)

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

の場合

:

分母と分子に以下を乗じる:

cos (x)

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

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

の場合

:

分母と分子に以下を乗じる:

sin (x)

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

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

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

= \frac{\frac{- c o s ^{2} ( x ) + s i n ^{2} ( x )}{s i n ( x ) c o s ( x )}}{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{s i n ( x ) c o s ( x )}}

分数を割る:

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

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

共通因数を約分する:

sin (x)

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

共通因数を約分する:

cos (x)

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

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

簡素化

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

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

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

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

条件のようなグループ

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

類似した元を足す:

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

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

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

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

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

条件のようなグループ

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

類似した元を足す:

cos^{2} (x) - cos^{2} (x) = 0

= 1

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

規則を適用

\frac{a}{1} = a

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

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

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

= 1 - 2 cos^{2} (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

証明する (sec^2(θ))/(sec^2(θ)-1)=csc^2(θ)

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

証明する cos(pi-θ)=-cos(θ)

p ro v e cos (π - θ) = - cos (θ)

証明する cos^4(θ)-sin^4(θ)=cos(2θ)

p ro v e cos^{4} (θ) - sin^{4} (θ) = cos (2 θ)

証明する sin(θ)-sin(θ)cos^2(θ)=sin^3(θ)

p ro v e sin (θ) - sin (θ) cos^{2} (θ) = sin^{3} (θ)

証明する csc^4(x)-cot^4(x)=2csc^2(x)-1

p ro v e csc^{4} (x) - cot^{4} (x) = 2 csc^{2} (x) - 1