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人気のある三角関数 >

証明する (1+cot^2(x))/(sec^2(x))=cot^2(x)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

サイン, コサインで表わす

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

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

指数の規則を適用する:

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

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

分数の規則を適用する:

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

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

結合

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

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

元を分数に変換する:

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

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

乗算:

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

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

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

分数の規則を適用する:

\frac{a}{1} = a

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

分数を乗じる:

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

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

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

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

ピタゴラスの公式を使用する:

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

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

簡素化

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

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

三角関数の公式を使用して書き換える

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

基本的な三角関数の公式を使用する:

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

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

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

基本的な三角関数の公式を使用する:

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

cot (x) cot (x)

簡素化

cot (x) cot (x) : cot^{2} (x)

cot (x) cot (x)

指数の規則を適用する:

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

cot (x) cot (x) = cot^{1 + 1} (x)

= cot^{1 + 1} (x)

数を足す:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

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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