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人気のある三角関数 >

証明する (cot^2(x)-1)/(2cot(x))=cot(2x)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

サイン, コサインで表わす

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

簡素化

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

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

乗じる

2 \cdot \frac{cos ( x )}{sin ( x )} : \frac{2 cos ( x )}{sin ( x )}

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

分数を乗じる:

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

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

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

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 ( x )}{2 cos ( x )}

乗じる

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

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

共通因数を約分する:

sin (x)

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

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

分数の規則を適用する:

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

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

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

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

2倍角の公式を使用:

2 sin (x) cos (x) = sin (2 x)

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

2倍角の公式を使用:

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

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

= \frac{cos ( 2 x )}{sin ( 2 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 (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(θ)=(sec(θ))/(tan(θ))

p ro v e csc (θ) = \frac{sec ( θ )}{tan ( θ )}

証明する (cos(2x)+1)/(sin(2x))=cot(x)

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

証明する-tan(t)-(cos(t))/(sin(t)-1)=sec(x)

p ro v e - tan (t) - \frac{cos ( t )}{sin ( t ) - 1} = sec (x)

証明する (cot(x))/(tan(x)sec(x))=csc(x)-1

p ro v e \frac{cot ( x )}{tan ( x ) sec ( x )} = csc (x) - 1

証明する cos(pi/2-x)cot(x)=cos(x)

p ro v e cos (\frac{π}{2} - x) cot (x) = cos (x)