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人気のある三角関数 >

証明する cot(2a)= 1/2 (cot(a)-tan(a))

解

証明する

cot (2 a) = \frac{1}{2} (cot (a) - tan (a))

解

真

解答ステップ

cot (2 a) = \frac{1}{2} (cot (a) - tan (a))

右側を操作する

\frac{1}{2} (cot (a) - tan (a))

サイン, コサインで表わす

(cot (a) - tan (a)) \frac{1}{2}

基本的な三角関数の公式を使用する:

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

= (\frac{cos ( a )}{sin ( a )} - tan (a)) \frac{1}{2}

基本的な三角関数の公式を使用する:

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

= (\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )}) \frac{1}{2}

簡素化

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

(\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )}) \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot ( \frac{c o s ( a )}{s i n ( a )} - \frac{s i n ( a )}{c o s ( a )} )}{2}

1 \cdot (\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )}) = \frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )}

1 \cdot (\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )})

乗算:

1 \cdot (\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )}) = (\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )})

= (\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )})

括弧を削除する:

(a) = a

= \frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )}

= \frac{\frac{c o s ( a )}{s i n ( a )} - \frac{s i n ( a )}{c o s ( a )}}{2}

結合

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

\frac{cos ( a )}{sin ( a )} - \frac{sin ( a )}{cos ( a )}

以下の最小公倍数:

sin (a), cos (a) : sin (a) cos (a)

sin (a), cos (a)

最小公倍数 (LCM)

sin (a)

または以下のいずれかに現れる因数で構成された式を計算する:

cos (a)

= sin (a) cos (a)

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

sin (a) cos (a)

\frac{cos ( a )}{sin ( a )}

の場合

:

分母と分子に以下を乗じる:

cos (a)

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

\frac{sin ( a )}{cos ( a )}

の場合

:

分母と分子に以下を乗じる:

sin (a)

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

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

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

= \frac{\frac{c o s ^{2} ( a ) - s i n ^{2} ( a )}{s i n ( a ) c o s ( a )}}{2}

分数の規則を適用する:

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

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

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

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

三角関数の公式を使用して書き換える

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

2倍角の公式を使用:

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

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

2倍角の公式を使用:

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

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

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

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 a)

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 (1-cos(2a))/(sin(2a))=tan(a)

p ro v e \frac{1 - cos ( 2 a )}{sin ( 2 a )} = tan (a)

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p ro v e sec (x) cos (- x) - sin^{2} (x) = cos^{2} (x)

証明する sin^4(x)=(1-cos^2(x))^2

p ro v e sin^{4} (x) = (1 - cos^{2} (x))^{2}

証明する 4/(cos^2(x))-5=4tan^2(x)-1

p ro v e \frac{4}{cos ^{2} ( x )} - 5 = 4 tan^{2} (x) - 1

証明する sin(x)+sin(y)=2sin((x+y)/2)cos((x-y)/2)

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