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人気のある三角関数 >

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

解

証明する

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

解

真

解答ステップ

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

右側を操作する

\frac{sec ^{2} ( θ ) csc ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

サイン, コサインで表わす

\frac{csc ^{2} ( θ ) sec ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

基本的な三角関数の公式を使用する:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2} sec ^{2} ( θ )}{( \frac{1}{s i n ( θ )} ) ^{2} - sec ^{2} ( θ )}

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

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

(\frac{1}{sin ( θ )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2} = \frac{1}{cos ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

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

(\frac{1}{sin ( θ )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

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

(\frac{1}{cos ( θ )})^{2} = \frac{1}{cos ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

結合

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

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

以下の最小公倍数:

sin^{2} (θ), cos^{2} (θ) : sin^{2} (θ) cos^{2} (θ)

sin^{2} (θ), cos^{2} (θ)

最小公倍数 (LCM)

sin^{2} (θ)

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^{2} (θ)

= sin^{2} (θ) cos^{2} (θ)

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^{2} (θ) cos^{2} (θ)

\frac{1}{sin ^{2} ( θ )}

の場合

:

分母と分子に以下を乗じる:

cos^{2} (θ)

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

\frac{1}{cos ^{2} ( θ )}

の場合

:

分母と分子に以下を乗じる:

sin^{2} (θ)

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

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

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} ( θ ) - sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

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

乗じる

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

\frac{1}{sin ^{2} ( θ )} \cdot \frac{1}{cos ^{2} ( θ )}

分数を乗じる:

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

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

数を乗じる:

1 \cdot 1 = 1

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

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

分数を割る:

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

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

改良

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

共通因数を約分する:

sin^{2} (θ)

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

共通因数を約分する:

cos^{2} (θ)

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

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

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

三角関数の公式を使用して書き換える

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

2倍角の公式を使用:

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

= \frac{1}{cos ( 2 θ )}

= \frac{1}{cos ( 2 θ )}

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 θ )}}

簡素化

\frac{1}{\frac{1}{s e c ( 2 θ )}}

分数の規則を適用する:

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

= \frac{sec ( 2 θ )}{1}

規則を適用

\frac{a}{1} = a

= sec (2 θ)

sec (2 θ)

sec (2 θ)

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 1/(1-sin(r))=sec^2(r)+sec(r)tan(r)

p ro v e \frac{1}{1 - sin ( r )} = sec^{2} (r) + sec (r) tan (r)

証明する 1/(sec(x))=sec(x)-tan(x)sin(x)

p ro v e \frac{1}{sec ( x )} = sec (x) - tan (x) sin (x)

証明する sin(a+b)sin(a-b)=sin^2(b)-sin^2(a)

p ro v e sin (a + b) sin (a - b) = sin^{2} (b) - sin^{2} (a)

証明する sec(b)+tan(b)=(cos(b))/(1-sin(b))

p ro v e sec (b) + tan (b) = \frac{cos ( b )}{1 - sin ( b )}

証明する cot^4(x)+cot^2(x)=csc^4(x)-csc^2(x)

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