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人気のある三角関数 >

証明する 1/(cot(x)+1)+1/(tan(x)+1)=1

解

証明する

\frac{1}{cot ( x ) + 1} + \frac{1}{tan ( x ) + 1} = 1

解

真

解答ステップ

\frac{1}{cot ( x ) + 1} + \frac{1}{tan ( x ) + 1} = 1

左側を操作する

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

サイン, コサインで表わす

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

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

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

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

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

結合

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

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

元を分数に変換する:

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

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

乗算:

1 \cdot sin (x) = sin (x)

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

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

分数の規則を適用する:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

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

分数の規則を適用する:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

規則を適用

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

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

規則を適用

\frac{a}{a} = 1

= 1

= 1

= 1

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(x)tan(x)=sec(x)

p ro v e csc (x) tan (x) = sec (x)

証明する 1+cot^2(x)= 1/(sin^2(x))

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

証明する tan(θ)cot(θ)-sin^2(θ)=cos^2(θ)

p ro v e tan (θ) cot (θ) - sin^{2} (θ) = cos^{2} (θ)

証明する sin(2x)=(2cot(x))/(1+cot^2(x))

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

証明する sec(θ)= 1/(cos(θ))

p ro v e sec (θ) = \frac{1}{cos ( θ )}