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人気のある三角関数 >

証明する cot(2b)=(cot^2(b)-1)/(2cot(b))

解

証明する

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

解

真

解答ステップ

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

右側を操作する

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

サイン, コサインで表わす

\frac{- 1 + cot ^{2} ( b )}{2 cot ( b )}

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 ( b )}{s i n ( b )} ) ^{2}}{2 \cdot \frac{c o s ( b )}{s i n ( b )}}

簡素化

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

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

乗じる

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

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

分数を乗じる:

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

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

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

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

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} ( b )}{s i n ^{2} ( b )} ) sin ( b )}{cos ( b ) \cdot 2}

結合

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

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

元を分数に変換する:

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

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

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} ( b ) + cos ^{2} ( b )}{sin ^{2} ( b )}

乗算:

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

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

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

乗じる

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

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

分数を乗じる:

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

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

共通因数を約分する:

sin (b)

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

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

分数の規則を適用する:

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

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

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

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

三角関数の公式を使用して書き換える

\frac{cos ^{2} ( b ) - sin ^{2} ( b )}{2 cos ( b ) sin ( b )}

2倍角の公式を使用:

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

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

2倍角の公式を使用:

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

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

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

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 b)

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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p ro v e 1 + tan^{2} (a) = sec^{2} (a)

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p ro v e 2 cos (x) cos (y) = cos (x + y) + cos (x - y)

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

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p ro v e cos (x + π) cos (x - π) = cos^{2} (x)

証明する (1+tan^2(u))(1-sin^2(u))=1

p ro v e (1 + tan^{2} (u)) (1 - sin^{2} (u)) = 1