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人気のある三角関数 >

証明する (csc^2(x))/(cot^2(x))=1+tan^2(x)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

\frac{csc ^{2} ( x )}{cot ^{2} ( x )}

サイン, コサインで表わす

\frac{csc ^{2} ( x )}{cot ^{2} ( x )}

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

簡素化

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

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

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

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

指数の規則を適用する:

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

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

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

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

分数を割る:

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

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

改良

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

共通因数を約分する:

sin^{2} (x)

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

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

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

右側を操作する

1 + tan^{2} (x)

サイン, コサインで表わす

1 + tan^{2} (x)

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

指数の規則を適用する:

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

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

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

ピタゴラスの公式を使用する:

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

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

= \frac{1}{cos ^{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

証明する sin(x/2)=sqrt((1+cos(x))/2)

p ro v e sin (\frac{x}{2}) = \frac{1 + cos ( x )}{2}

証明する 1+1/(tan^2(θ))= 1/(sin^2(θ))

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

証明する sin^2(x)*(1+cot^2(x))=1

p ro v e sin^{2} (x) \cdot (1 + cot^{2} (x)) = 1

証明する cos(x+1)=cos(-x-1)

p ro v e cos (x + 1) = cos (- x - 1)

証明する cot^2(x)+1= 1/(sin^2(x))

p ro v e cot^{2} (x) + 1 = \frac{1}{sin ^{2} ( x )}