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人気のある三角関数 >

証明する (sec(a))/(cot(a)+tan(a))=sin(a)

解

証明する

\frac{sec ( a )}{cot ( a ) + tan ( a )} = sin (a)

解

真

解答ステップ

\frac{sec ( a )}{cot ( a ) + tan ( a )} = sin (a)

左側を操作する

\frac{sec ( a )}{cot ( a ) + tan ( a )}

サイン, コサインで表わす

\frac{sec ( a )}{cot ( a ) + tan ( a )}

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 ( a )}}{cot ( a ) + tan ( a )}

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 ( a )}}{\frac{c o s ( a )}{s i n ( a )} + tan ( a )}

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 ( a )}}{\frac{c o s ( a )}{s i n ( a )} + \frac{s i n ( a )}{c o s ( a )}}

簡素化

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

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

分数の規則を適用する:

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

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

結合

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

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

以下の最小公倍数:

sin (a), cos (a) : sin (a) cos (a)

sin (a), cos (a)

最小公倍数 (LCM)

sin (a)

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 (a)

= sin (a) cos (a)

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 (a) cos (a)

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

の場合

:

分母と分子に以下を乗じる:

cos (a)

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

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

の場合

:

分母と分子に以下を乗じる:

sin (a)

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

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

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

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

乗じる

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (a)

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

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

分数の規則を適用する:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

= \frac{sin ( a )}{1}

規則を適用

\frac{a}{1} = a

= sin (a)

= sin (a)

両辺を同じ形式にできることを証明した

\Rightarrow 真

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