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人気のある三角関数 >

証明する sec^2(B)-csc^2(B)=(tan(B)-cot(B))/(sin(B)cos(B))

解

証明する

sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

解

真

解答ステップ

sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

右側を操作する

\frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

サイン, コサインで表わす

\frac{- cot ( B ) + tan ( B )}{cos ( B ) sin ( B )}

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 ( B )}{s i n ( B )} + tan ( B )}{cos ( B ) sin ( B )}

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 ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )}

簡素化

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

\frac{- \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )}

結合

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

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

以下の最小公倍数:

sin (B), cos (B) : sin (B) cos (B)

sin (B), cos (B)

最小公倍数 (LCM)

sin (B)

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 (B)

= sin (B) cos (B)

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 (B) cos (B)

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

の場合

:

分母と分子に以下を乗じる:

cos (B)

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

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

の場合

:

分母と分子に以下を乗じる:

sin (B)

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

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

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

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

分数の規則を適用する:

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

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

sin (B) cos (B) cos (B) sin (B) = cos^{2} (B) sin^{2} (B)

sin (B) cos (B) cos (B) sin (B)

指数の規則を適用する:

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

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

= sin (B) cos^{1 + 1} (B) sin (B)

数を足す:

1 + 1 = 2

= sin (B) cos^{2} (B) sin (B)

指数の規則を適用する:

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

sin (B) sin (B) = sin^{1 + 1} (B)

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

数を足す:

1 + 1 = 2

= cos^{2} (B) sin^{2} (B)

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

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

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

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:

sin (x) = \frac{1}{csc ( x )}

\frac{- cos ^{2} ( B ) + ( \frac{1}{c s c ( B )} ) ^{2}}{cos ^{2} ( B ) ( \frac{1}{c s c ( B )} ) ^{2}}

基本的な三角関数の公式を使用する:

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

\frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{s e c ( B )} ) ^{2} ( \frac{1}{c s c ( B )} ) ^{2}}

簡素化

\frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{s e c ( B )} ) ^{2} ( \frac{1}{c s c ( B )} ) ^{2}}

(\frac{1}{sec ( B )})^{2} = \frac{1}{sec ^{2} ( B )}

(\frac{1}{sec ( B )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{sec ^{2} ( B )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( B )}

= \frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{c s c ( B )} ) ^{2} \frac{1}{s e c ^{2} ( B )}}

(\frac{1}{csc ( B )})^{2} = \frac{1}{csc ^{2} ( B )}

(\frac{1}{csc ( B )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{csc ^{2} ( B )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( B )}

= \frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{\frac{1}{s e c ^{2} ( B )} \cdot \frac{1}{c s c ^{2} ( B )}}

(\frac{1}{sec ( B )})^{2} = \frac{1}{sec ^{2} ( B )}

(\frac{1}{sec ( B )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{sec ^{2} ( B )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( B )}

(\frac{1}{csc ( B )})^{2} = \frac{1}{csc ^{2} ( B )}

(\frac{1}{csc ( B )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{csc ^{2} ( B )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( B )}

= \frac{- \frac{1}{s e c ^{2} ( B )} + \frac{1}{c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B )} \cdot \frac{1}{c s c ^{2} ( B )}}

乗じる

\frac{1}{sec ^{2} ( B )} \cdot \frac{1}{csc ^{2} ( B )} : \frac{1}{sec ^{2} ( B ) csc ^{2} ( B )}

\frac{1}{sec ^{2} ( B )} \cdot \frac{1}{csc ^{2} ( B )}

分数を乗じる:

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

= \frac{1 \cdot 1}{sec ^{2} ( B ) csc ^{2} ( B )}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{sec ^{2} ( B ) csc ^{2} ( B )}

= \frac{- \frac{1}{s e c ^{2} ( B )} + \frac{1}{c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B ) c s c ^{2} ( B )}}

結合

- \frac{1}{sec ^{2} ( B )} + \frac{1}{csc ^{2} ( B )} : \frac{- csc ^{2} ( B ) + sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

- \frac{1}{sec ^{2} ( B )} + \frac{1}{csc ^{2} ( B )}

以下の最小公倍数:

sec^{2} (B), csc^{2} (B) : sec^{2} (B) csc^{2} (B)

sec^{2} (B), csc^{2} (B)

最小公倍数 (LCM)

sec^{2} (B)

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:

csc^{2} (B)

= sec^{2} (B) csc^{2} (B)

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

sec^{2} (B) csc^{2} (B)

\frac{1}{sec ^{2} ( B )}

の場合

:

分母と分子に以下を乗じる:

csc^{2} (B)

\frac{1}{sec ^{2} ( B )} = \frac{1 \cdot csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )} = \frac{csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

\frac{1}{csc ^{2} ( B )}

の場合

:

分母と分子に以下を乗じる:

sec^{2} (B)

\frac{1}{csc ^{2} ( B )} = \frac{1 \cdot sec ^{2} ( B )}{csc ^{2} ( B ) sec ^{2} ( B )} = \frac{sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

= - \frac{csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )} + \frac{sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

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{- csc ^{2} ( B ) + sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

= \frac{\frac{- c s c ^{2} ( B ) + s e c ^{2} ( B )}{s e c ^{2} ( B ) c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B ) c s c ^{2} ( B )}}

分数を割る:

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

= \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) sec ^{2} ( B ) csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B ) \cdot 1}

改良

= \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) sec ^{2} ( B ) csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

共通因数を約分する:

sec^{2} (B)

= - \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) csc ^{2} ( B )}{csc ^{2} ( B )}

共通因数を約分する:

csc^{2} (B)

= - - csc^{2} (B) + sec^{2} (B)

- csc^{2} (B) + sec^{2} (B)

- csc^{2} (B) + sec^{2} (B)

= sec^{2} (B) - csc^{2} (B)

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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