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人気のある三角関数 >

証明する (tan^2(x))/(sec(x))=sec(x)-cos(x)

解

証明する

\frac{tan ^{2} ( x )}{sec ( x )} = sec (x) - cos (x)

解

真

解答ステップ

\frac{tan ^{2} ( x )}{sec ( x )} = sec (x) - cos (x)

左側を操作する

\frac{tan ^{2} ( x )}{sec ( x )}

サイン, コサインで表わす

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

簡素化

\frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{\frac{1}{c o s ( x )}} : \frac{sin ^{2} ( x )}{cos ( x )}

\frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{\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{( \frac{s i n ( x )}{c o s ( x )} ) ^{2} cos ( x )}{1}

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

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

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

乗じる

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

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

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

分数の規則を適用する:

\frac{a}{1} = a

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

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

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

三角関数の公式を使用して書き換える

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

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

右側を操作する

sec (x) - cos (x)

サイン, コサインで表わす

- cos (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 )}

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

簡素化

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

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

元を分数に変換する:

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

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

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

- cos (x) cos (x) + 1

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

cos (x) cos (x)

指数の規則を適用する:

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

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

= cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= cos^{2} (x)

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

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

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

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

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

証明する tanh(x)=(sinh(x))/(cosh(x))

p ro v e tanh (x) = \frac{sinh ( x )}{cosh ( x )}

証明する sin(x)sec(x)=tan(x)

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

証明する cos(4x)=8cos^4(x)-8cos^2(x)+1

p ro v e cos (4 x) = 8 cos^{4} (x) - 8 cos^{2} (x) + 1

証明する cos(2x)=2cos^2(x)-1

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