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人気のある三角関数 >

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

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

三角関数の公式を使用して書き換える

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

以下を乗じる:

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

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

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

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

= \frac{sec ( x )}{\frac{1 - c o s ^{2} ( x )}{1 + c o s ( x )}}

ピタゴラスの公式を使用する:

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

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

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

分数の規則を適用する:

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

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

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

拡張

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

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

拡張

(1 + cos (x)) sec (x) : sec (x) + sec (x) cos (x)

(1 + cos (x)) sec (x)

= sec (x) (1 + cos (x))

分配法則を適用する:

a (b + c) = ab + a c

a = sec (x), b = 1, c = cos (x)

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

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

乗算:

1 \cdot sec (x) = sec (x)

= sec (x) + sec (x) cos (x)

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

分数の規則を適用する:

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

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

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

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

三角関数の公式を使用して書き換える

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

sec (x) cos (x) = 1

sec (x) cos (x)

サイン, コサインで表わす

sec (x) cos (x)

基本的な三角関数の公式を使用する:

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

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

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

= 1

= 1

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

規則を適用

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

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

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

= \frac{sec ( x ) + 1}{sin ^{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 tan(pi/4-θ)=(cos(θ)-sin(θ))/(cos(θ)+sin(θ))

p ro v e tan (\frac{π}{4} - θ) = \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

証明する cot(2θ)= 1/2 sec(θ)csc(θ)-tan(θ)

p ro v e cot (2 θ) = \frac{1}{2} sec (θ) csc (θ) - tan (θ)

証明する 8cos^2(x)-8sin^2(x)=8-16sin^2(x)

p ro v e 8 cos^{2} (x) - 8 sin^{2} (x) = 8 - 16 sin^{2} (x)

証明する sec(x)-sin(x)*tan(x)=cos(x)

p ro v e sec (x) - sin (x) \cdot tan (x) = cos (x)

証明する (sec(x)csc(x))/(cot(x))=sec^2(x)

p ro v e \frac{sec ( x ) csc ( x )}{cot ( x )} = sec^{2} (x)