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人気のある三角関数 >

証明する sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t)

解

証明する

sec (t) (csc (t) (tan (t) + cot (t))) = sec^{2} (t) + csc^{2} (t)

解

真

解答ステップ

sec (t) csc (t) (tan (t) + cot (t)) = sec^{2} (t) + csc^{2} (t)

左側を操作する

sec (t) csc (t) (tan (t) + cot (t))

サイン, コサインで表わす

(cot (t) + tan (t)) csc (t) sec (t)

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 ( t )}{sin ( t )} + tan (t)) csc (t) sec (t)

基本的な三角関数の公式を使用する:

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

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) csc (t) sec (t)

基本的な三角関数の公式を使用する:

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

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} sec (t)

基本的な三角関数の公式を使用する:

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

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )}

簡素化

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

(\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )}

分数を乗じる:

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

= \frac{1 \cdot 1 \cdot ( \frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )} )}{sin ( t ) cos ( t )}

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) = \frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )})

乗算:

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) = (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )})

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )})

括弧を削除する:

(a) = a

= \frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

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

結合

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

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

以下の最小公倍数:

sin (t), cos (t) : sin (t) cos (t)

sin (t), cos (t)

最小公倍数 (LCM)

sin (t)

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 (t)

= sin (t) cos (t)

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 (t) cos (t)

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

の場合

:

分母と分子に以下を乗じる:

cos (t)

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

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

の場合

:

分母と分子に以下を乗じる:

sin (t)

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

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

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

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

分数の規則を適用する:

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

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

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

sin (t) cos (t) sin (t) cos (t)

指数の規則を適用する:

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

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

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

数を足す:

1 + 1 = 2

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

指数の規則を適用する:

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

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

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

数を足す:

1 + 1 = 2

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

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

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

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

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} ( t ) + ( \frac{1}{c s c ( t )} ) ^{2}}{cos ^{2} ( t ) ( \frac{1}{c s c ( t )} ) ^{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 ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{( \frac{1}{s e c ( t )} ) ^{2} ( \frac{1}{c s c ( t )} ) ^{2}}

簡素化

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

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

(\frac{1}{sec ( t )})^{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} ( t )}

規則を適用

1^{a} = 1

1^{2} = 1

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

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

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

(\frac{1}{csc ( t )})^{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} ( t )}

規則を適用

1^{a} = 1

1^{2} = 1

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

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

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

(\frac{1}{sec ( t )})^{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} ( t )}

規則を適用

1^{a} = 1

1^{2} = 1

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

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

(\frac{1}{csc ( t )})^{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} ( t )}

規則を適用

1^{a} = 1

1^{2} = 1

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

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

乗じる

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

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

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} ( t ) csc ^{2} ( t )}

数を乗じる:

1 \cdot 1 = 1

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

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

結合

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

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

以下の最小公倍数:

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

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

最小公倍数 (LCM)

sec^{2} (t)

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} (t)

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

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} (t) csc^{2} (t)

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

の場合

:

分母と分子に以下を乗じる:

csc^{2} (t)

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

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

の場合

:

分母と分子に以下を乗じる:

sec^{2} (t)

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

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

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} ( t ) + sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

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

分数を割る:

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

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

改良

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

共通因数を約分する:

sec^{2} (t)

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

共通因数を約分する:

csc^{2} (t)

= csc^{2} (t) + sec^{2} (t)

csc^{2} (t) + sec^{2} (t)

csc^{2} (t) + sec^{2} (t)

= sec^{2} (t) + csc^{2} (t)

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

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p ro v e cot (6 0^{\circ}) = \frac{cos ( 6 0 ^{\circ} )}{sin ( 6 0 ^{\circ} )}

証明する tan(-x)tan(pi/2-x)=-1

p ro v e tan (- x) tan (\frac{π}{2} - x) = - 1

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p ro v e tan (π - θ) = - tan (x)

証明する cot(θ)(sin(θ)+tan(θ))=cos(θ)+1

p ro v e cot (θ) (sin (θ) + tan (θ)) = cos (θ) + 1