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人気のある三角関数 >

証明する (cot^2(x)-1)/(csc^2(x))=cos(2x)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

\frac{cot ^{2} ( x ) - 1}{csc ^{2} ( x )}

サイン, コサインで表わす

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

簡素化

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

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

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

共通因数を約分する:

sin^{2} (x)

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

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

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

2倍角の公式を使用:

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

= cos (2 x)

= 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 csc^2(x)-csc^2(x)*cos^2(x)=1

p ro v e csc^{2} (x) - csc^{2} (x) \cdot cos^{2} (x) = 1

証明する cos(θ)-sin(θ)sin(2θ)=cos(θ)cos(2θ)

p ro v e cos (θ) - sin (θ) sin (2 θ) = cos (θ) cos (2 θ)

証明する 1+(cot(A))^2=*(csc(A))^2

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

証明する (sin(x)*cos(x))/(tan(x))=cos^2(x)

p ro v e \frac{sin ( x ) \cdot cos ( x )}{tan ( x )} = cos^{2} (x)

証明する 1/(cos^2(x))=20

p ro v e \frac{1}{cos ^{2} ( x )} = 20