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人気のある三角関数 >

証明する (tan(x)+cot(x))/(sec(x)csc(x))=1

解

証明する

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

解

真

解答ステップ

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

左側を操作する

\frac{tan ( x ) + cot ( x )}{sec ( x ) csc ( x )}

サイン, コサインで表わす

\frac{cot ( x ) + tan ( x )}{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{\frac{c o s ( x )}{s i n ( x )} + tan ( x )}{csc ( 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{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{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{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\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{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{s i n ( x )} \cdot \frac{1}{c o s ( x )}}

簡素化

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

\frac{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{s i n ( x )} \cdot \frac{1}{c o s ( x )}}

乗じる

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{1}{sin ( x ) cos ( x )}

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

分数を乗じる:

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

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

数を乗じる:

1 \cdot 1 = 1

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

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

結合

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

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

以下の最小公倍数:

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

sin (x)

または以下のいずれかに現れる因数で構成された式を計算する:

cos (x)

= sin (x) cos (x)

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

sin (x) cos (x)

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

の場合

:

分母と分子に以下を乗じる:

cos (x)

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

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

の場合

:

分母と分子に以下を乗じる:

sin (x)

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

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

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

分数を割る:

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

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

改良

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

共通因数を約分する:

sin (x)

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

共通因数を約分する:

cos (x)

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 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 tan^2(a)+sin^2(a)=tan^2(a)sin^2(a)

p ro v e tan^{2} (a) + sin^{2} (a) = tan^{2} (a) sin^{2} (a)

証明する 1/(1-sin(x))=-1/(sin(x))

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

証明する tan(-a)=-tan(a)

p ro v e tan (- a) = - tan (a)

証明する (tan(y)+cos(y))/(1+sin(y))=sec(y)

p ro v e \frac{tan ( y ) + cos ( y )}{1 + sin ( y )} = sec (y)

証明する 4sin(3x)cos(4x)=2sin(7x)-2sin(x)

p ro v e 4 sin (3 x) cos (4 x) = 2 sin (7 x) - 2 sin (x)