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人気のある三角関数 >

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

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

サイン, コサインで表わす

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

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 ( t )}{c o s ( t )} ) ^{2}}{sec ( t )}

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 ( t )}{c o s ( t )} ) ^{2}}{\frac{1}{c o s ( t )}}

簡素化

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

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

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 ( t )}{c o s ( t )} ) ^{2} cos ( t )}{1}

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

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

指数の規則を適用する:

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

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

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

乗じる

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (t)

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

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

分数の規則を適用する:

\frac{a}{1} = a

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

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

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

三角関数の公式を使用して書き換える

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

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} ( t )}{cos ( t )}

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

右側を操作する

sec (t) - cos (t)

サイン, コサインで表わす

- cos (t) + sec (t)

基本的な三角関数の公式を使用する:

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

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

簡素化

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

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

元を分数に変換する:

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

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

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 ( t ) cos ( t ) + 1}{cos ( t )}

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

- cos (t) cos (t) + 1

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

cos (t) cos (t)

指数の規則を適用する:

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

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

= cos^{1 + 1} (t)

数を足す:

1 + 1 = 2

= cos^{2} (t)

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

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

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

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

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

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p ro v e - sin (- 1) = sin (1)

証明する sin^2(y)cos^2(y)=(1-cos(4x))/8

p ro v e sin^{2} (y) cos^{2} (y) = \frac{1 - cos ( 4 x )}{8}

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

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

証明する ((cos(a-b)))/(sin(a)sin(b))=cot(a)cot(b)+1

p ro v e \frac{( cos ( a - b ) )}{sin ( a ) sin ( b )} = cot (a) cot (b) + 1

証明する 2cos(x)(sin(3x)-sin(x))=sin(4x)

p ro v e 2 cos (x) (sin (3 x) - sin (x)) = sin (4 x)