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人気のある三角関数 >

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

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

サイン, コサインで表わす

\frac{1 + cot ^{2} ( x )}{- 1 + cot ^{2} ( 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{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{- 1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

簡素化

\frac{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{- 1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} : \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{- sin ^{2} ( x ) + cos ^{2} ( x )}

\frac{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{- 1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

指数の規則を適用する:

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

= \frac{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{- 1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

指数の規則を適用する:

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

= \frac{1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}{- 1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

結合

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

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

元を分数に変換する:

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

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

乗算:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

= \frac{1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}{\frac{- s i n ^{2} ( x ) + c o s ^{2} ( x )}{s i n ^{2} ( x )}}

結合

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

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

元を分数に変換する:

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

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

乗算:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

分数を割る:

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

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

共通因数を約分する:

sin^{2} (x)

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

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

2倍角の公式を使用:

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

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

ピタゴラスの公式を使用する:

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

= \frac{1}{cos ( 2 x )}

= \frac{1}{cos ( 2 x )}

三角関数の公式を使用して書き換える

基本的な三角関数の公式を使用する:

cos (x) = \frac{1}{sec ( x )}

\frac{1}{\frac{1}{s e c ( 2 x )}}

簡素化

\frac{1}{\frac{1}{s e c ( 2 x )}}

分数の規則を適用する:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ( 2 x )}{1}

規則を適用

\frac{a}{1} = a

= sec (2 x)

sec (2 x)

sec (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

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

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