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人気のある三角関数 >

証明する (1+tan^2(θ))/(tan(θ))=sec(θ)csc(θ)

解

証明する

\frac{1 + tan ^{2} ( θ )}{tan ( θ )} = sec (θ) csc (θ)

解

真

解答ステップ

\frac{1 + tan ^{2} ( θ )}{tan ( θ )} = sec (θ) csc (θ)

左側を操作する

\frac{1 + tan ^{2} ( θ )}{tan ( θ )}

サイン, コサインで表わす

\frac{1 + tan ^{2} ( θ )}{tan ( θ )}

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

分数の規則を適用する:

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

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

指数の規則を適用する:

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

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

結合

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

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

元を分数に変換する:

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

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

乗算:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

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

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

乗じる

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

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ )} cos (θ)

分数を乗じる:

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

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

共通因数を約分する:

cos (θ)

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

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

分数の規則を適用する:

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

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

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

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

三角関数の公式を使用して書き換える

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ( θ ) sin ( θ )}

ピタゴラスの公式を使用する:

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

= \frac{1}{cos ( θ ) sin ( θ )}

= \frac{1}{cos ( θ ) sin ( 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:

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

\frac{1}{cos ( θ ) \frac{1}{c s c ( θ )}}

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 ( θ )} \cdot \frac{1}{c s c ( θ )}}

簡素化

\frac{1}{\frac{1}{s e c ( θ )} \cdot \frac{1}{c s c ( θ )}}

乗じる

\frac{1}{sec ( θ )} \cdot \frac{1}{csc ( θ )} : \frac{1}{sec ( θ ) csc ( θ )}

\frac{1}{sec ( θ )} \cdot \frac{1}{csc ( 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}{sec ( θ ) csc ( θ )}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{sec ( θ ) csc ( θ )}

= \frac{1}{\frac{1}{s e c ( θ ) c s c ( θ )}}

分数の規則を適用する:

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

= \frac{sec ( θ ) csc ( θ )}{1}

規則を適用

\frac{a}{1} = a

= sec (θ) csc (θ)

sec (θ) csc (θ)

sec (θ) csc (θ)

両辺を同じ形式にできることを証明した

\Rightarrow 真

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