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人気のある三角関数 >

証明する (sec(t))/(cos(t))-(tan(t))/(cot(t))=1

解

証明する

\frac{sec ( t )}{cos ( t )} - \frac{tan ( t )}{cot ( t )} = 1

解

真

解答ステップ

\frac{sec ( t )}{cos ( t )} - \frac{tan ( t )}{cot ( t )} = 1

左側を操作する

\frac{sec ( t )}{cos ( t )} - \frac{tan ( t )}{cot ( t )}

サイン, コサインで表わす

\frac{sec ( t )}{cos ( t )} - \frac{tan ( t )}{cot ( 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{1}{c o s ( t )}}{cos ( t )} - \frac{tan ( t )}{cot ( 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{1}{c o s ( t )}}{cos ( t )} - \frac{\frac{s i n ( t )}{c o s ( t )}}{cot ( t )}

基本的な三角関数の公式を使用する:

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

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

簡素化

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

\frac{\frac{1}{c o s ( t )}}{cos ( t )} - \frac{\frac{s i n ( t )}{c o s ( t )}}{\frac{c o s ( t )}{s i n ( t )}}

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

\frac{\frac{1}{c o s ( t )}}{cos ( t )}

分数の規則を適用する:

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

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

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)

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

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

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

分数を割る:

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

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

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

sin (t) sin (t)

指数の規則を適用する:

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

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

= sin^{1 + 1} (t)

数を足す:

1 + 1 = 2

= sin^{2} (t)

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

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)

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

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

規則を適用

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

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

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

規則を適用

\frac{a}{a} = 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 (cot(y))/(sec(y))=csc(y)-sin(y)

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

証明する csc(θ)=cos(θ)cot(θ)+sin(θ)

p ro v e csc (θ) = cos (θ) cot (θ) + sin (θ)

証明する sin(z+pi/2)=cos(z)

p ro v e sin (z + \frac{π}{2}) = cos (z)

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

p ro v e \frac{sec ( α )}{1 - cos ( α )} = \frac{sec ( α ) + 1}{sin ^{2} ( α )}

証明する cos(pi/4)=(sqrt(2))/2

p ro v e cos (\frac{π}{4}) = \frac{2}{2}