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人気のある三角関数 >

証明する (1+csc(x))/(sec(x))-cot(x)=cos(x)

解

証明する

\frac{1 + csc ( x )}{sec ( x )} - cot (x) = cos (x)

解

真

解答ステップ

\frac{1 + csc ( x )}{sec ( x )} - cot (x) = cos (x)

左側を操作する

\frac{1 + csc ( x )}{sec ( x )} - cot (x)

サイン, コサインで表わす

- cot (x) + \frac{1 + csc ( x )}{sec ( 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{cos ( x )}{sin ( x )} + \frac{1 + csc ( x )}{sec ( x )}

基本的な三角関数の公式を使用する:

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

= - \frac{cos ( x )}{sin ( x )} + \frac{1 + \frac{1}{s i n ( x )}}{sec ( x )}

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

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

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

結合

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

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

乗算:

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

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

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

分数の規則を適用する:

\frac{a}{1} = a

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

分数を乗じる:

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

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

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

規則を適用

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

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

因数

- cos (x) + (sin (x) + 1) cos (x) : cos (x) sin (x)

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

共通項をくくり出す

cos (x)

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

改良

= cos (x) sin (x)

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

共通因数を約分する:

sin (x)

= cos (x)

= cos (x)

= cos (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 1/(1+cos(x))+1/(1-cos(x))=2csc^2(x)

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

証明する (csc(-x))/(sec(-x))=-cot(x)

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

証明する cos(x-y)+cos(x+y)=2cos(x)cos(y)

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

証明する sin(pi/6+x)+sin(pi/6-x)=cos(x)

p ro v e sin (\frac{π}{6} + x) + sin (\frac{π}{6} - x) = cos (x)

証明する (cot(x)-csc(x))(cos(x)+1)=-sin(x)

p ro v e (cot (x) - csc (x)) (cos (x) + 1) = - sin (x)