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人気のある三角関数 >

証明する csc(t)cot(t)=(1+cot^2(t))/(sec(t))

解

証明する

csc (t) cot (t) = \frac{1 + cot ^{2} ( t )}{sec ( t )}

解

真

解答ステップ

csc (t) cot (t) = \frac{1 + cot ^{2} ( t )}{sec ( t )}

左側を操作する

csc (t) cot (t)

サイン, コサインで表わす

cot (t) csc (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{cos ( t )}{sin ( t )} csc (t)

基本的な三角関数の公式を使用する:

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

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

簡素化

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

\frac{cos ( t )}{sin ( t )} \cdot \frac{1}{sin ( t )}

分数を乗じる:

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

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

乗算:

cos (t) \cdot 1 = cos (t)

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

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

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

右側を操作する

\frac{1 + cot ^{2} ( t )}{sec ( t )}

サイン, コサインで表わす

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

簡素化

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

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

指数の規則を適用する:

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

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

結合

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

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

元を分数に変換する:

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

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

乗算:

1 \cdot sin^{2} (t) = sin^{2} (t)

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

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

分数の規則を適用する:

\frac{a}{1} = a

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

簡素化

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

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

証明する csc(a)sec(a)=2csc(2a)

p ro v e csc (a) sec (a) = 2 csc (2 a)

証明する tan^2(2x)+1=sec^2(2x)

p ro v e tan^{2} (2 x) + 1 = sec^{2} (2 x)

証明する sin(4θ)=4sin(θ)cos(θ)cos(2θ)

p ro v e sin (4 θ) = 4 sin (θ) cos (θ) cos (2 θ)

証明する sin^9(x)=(1-cos^2(x))^4sin(x)

p ro v e sin^{9} (x) = (1 - cos^{2} (x))^{4} sin (x)

証明する csc^2(θ)=1-cot^2(θ)

p ro v e csc^{2} (θ) = 1 - cot^{2} (θ)