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人気のある三角関数 >

証明する (1-tan(x))/(1+tan(x))=(1-sin(2x))/(cos(2x))

解

証明する

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

解

真

解答ステップ

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

左側を操作する

\frac{1 - tan ( x )}{1 + tan ( x )}

サイン, コサインで表わす

\frac{1 - tan ( x )}{1 + tan ( x )}

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

結合

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

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

元を分数に変換する:

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

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

乗算:

1 \cdot cos (x) = cos (x)

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

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

結合

1 - \frac{sin ( x )}{cos ( x )} : \frac{cos ( x ) - sin ( x )}{cos ( x )}

1 - \frac{sin ( x )}{cos ( x )}

元を分数に変換する:

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

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

乗算:

1 \cdot cos (x) = cos (x)

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

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

分数を割る:

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

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

共通因数を約分する:

cos (x)

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

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

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

以下を乗じる:

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

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

拡張

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

(cos (x) - sin (x)) (cos (x) - sin (x)) cos (2 x)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

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

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

数を足す:

1 + 1 = 2

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

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

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

完全平方式を適用する:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = cos (x), b = sin (x)

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

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

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

括弧を分配する

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

マイナス・プラスの規則を適用する

+ (- a) = - a

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

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

ピタゴラスの公式を使用する:

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

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

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

簡素化

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

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

因数

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

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

共通項をくくり出す

cos (2 x)

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

改良

= cos (2 x) (- 2 cos (x) sin (x) + 1)

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

共通因数を約分する:

cos (2 x)

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

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

2倍角の公式を使用:

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

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

拡張

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

(cos (x) + sin (x)) (cos (x) - sin (x))

2乗の差の公式を適用する:

(a + b) (a - b) = a^{2} - b^{2}

a = cos (x), b = sin (x)

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

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

2倍角の公式を使用:

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

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

= \frac{1 - sin ( 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 csch^2(x)=coth^2(x)-1

p ro v e csch^{2} (x) = coth^{2} (x) - 1

証明する (tan(y))/(csc(y))= 1/(cos(y))-1/(sec(y))

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

証明する 2tan(x)sec(x)= 1/(1-sin(x))-1/(1+sin(x))

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

証明する 1/(sin(x)cot(x))= 1/(cos(x))

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

証明する (csc^2(x)-1)/(csc^2(x))=cos^2(x)

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