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人気のある三角関数 >

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

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

サイン, コサインで表わす

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

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 ( x )}{c o s ( x )} ) ^{2}}{csc ( x ) sec ( x )}

基本的な三角関数の公式を使用する:

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

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

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

乗じる

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{1}{sin ( x ) cos ( x )}

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )}

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 ( x ) cos ( x )}

数を乗じる:

1 \cdot 1 = 1

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

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

指数の規則を適用する:

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

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

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 ^{2} ( x )}{c o s ^{2} ( x )} ) sin ( x ) cos ( x )}{1}

結合

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

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

元を分数に変換する:

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

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

乗算:

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

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

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

分数の規則を適用する:

\frac{a}{1} = a

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

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

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

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

ピタゴラスの公式を使用する:

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

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

簡素化

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

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

基本的な三角関数の公式を使用する:

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

= tan (x)

両辺を同じ形式にできることを証明した

\Rightarrow 真

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