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人気のある三角関数 >

証明する (cos^4(x)-sin^4(x))/(cos^2(x))=1-tan^2(x)

解

証明する

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

解

真

解答ステップ

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

左側を操作する

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

三角関数の公式を使用して書き換える

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

ピタゴラスの公式を使用する:

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

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

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

右側を操作する

1 - tan^{2} (x)

三角関数の公式を使用して書き換える

1 - tan^{2} (x)

ピタゴラスの公式を使用する:

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

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

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

サイン, コサインで表わす

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

基本的な三角関数の公式を使用する:

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

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

簡素化

cos^{2} (x) + sin^{2} (x) - (\frac{sin ( x )}{cos ( x )})^{2} : \frac{cos ^{4} ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

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

指数の規則を適用する:

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

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

元を分数に変換する:

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

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

cos^{2} (x) cos^{2} (x) + sin^{2} (x) cos^{2} (x) - sin^{2} (x) = cos^{4} (x) + sin^{2} (x) cos^{2} (x) - sin^{2} (x)

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

cos^{2} (x) cos^{2} (x) = cos^{4} (x)

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

指数の規則を適用する:

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

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

= cos^{2 + 2} (x)

数を足す:

2 + 2 = 4

= cos^{4} (x)

= cos^{4} (x) + sin^{2} (x) cos^{2} (x) - sin^{2} (x)

= \frac{cos ^{4} ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{4} ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{4} ( x ) - sin ^{2} ( x ) + cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x )}

三角関数の公式を使用して書き換える

\frac{cos ^{4} ( x ) - sin ^{2} ( x ) + cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x )}

ピタゴラスの公式を使用する:

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

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

= \frac{cos ^{4} ( x ) - sin ^{2} ( x ) + ( 1 - sin ^{2} ( x ) ) sin ^{2} ( x )}{1 - sin ^{2} ( x )}

拡張

cos^{4} (x) - sin^{2} (x) + (1 - sin^{2} (x)) sin^{2} (x) : cos^{4} (x) - sin^{4} (x)

cos^{4} (x) - sin^{2} (x) + (1 - sin^{2} (x)) sin^{2} (x)

= cos^{4} (x) - sin^{2} (x) + sin^{2} (x) (1 - sin^{2} (x))

拡張

sin^{2} (x) (1 - sin^{2} (x)) : sin^{2} (x) - sin^{4} (x)

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

分配法則を適用する:

a (b - c) = ab - a c

a = sin^{2} (x), b = 1, c = sin^{2} (x)

= sin^{2} (x) \cdot 1 - sin^{2} (x) sin^{2} (x)

= 1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x)

簡素化

1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x) : sin^{2} (x) - sin^{4} (x)

1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x)

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

1 \cdot sin^{2} (x)

乗算:

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

= sin^{2} (x)

sin^{2} (x) sin^{2} (x) = sin^{4} (x)

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

指数の規則を適用する:

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

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

= sin^{2 + 2} (x)

数を足す:

2 + 2 = 4

= sin^{4} (x)

= sin^{2} (x) - sin^{4} (x)

= sin^{2} (x) - sin^{4} (x)

= cos^{4} (x) - sin^{2} (x) + sin^{2} (x) - sin^{4} (x)

類似した元を足す:

- sin^{2} (x) + sin^{2} (x) = 0

= cos^{4} (x) - sin^{4} (x)

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

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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