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人気のある三角関数 >

証明する 4/(cos^2(x))-5=4tan^2(x)-1

解

証明する

\frac{4}{cos ^{2} ( x )} - 5 = 4 tan^{2} (x) - 1

解

真

解答ステップ

\frac{4}{cos ^{2} ( x )} - 5 = 4 tan^{2} (x) - 1

右側を操作する

4 tan^{2} (x) - 1

サイン, コサインで表わす

- 1 + 4 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 )}

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

簡素化

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

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

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

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

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

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

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

分数を乗じる:

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

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

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

元を分数に変換する:

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

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

乗算:

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

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

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

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

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

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

拡張

- cos^{2} (x) + 4 (1 - cos^{2} (x)) : - 5 cos^{2} (x) + 4

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

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (x)

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

数を乗じる:

4 \cdot 1 = 4

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

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

簡素化

- cos^{2} (x) + 4 - 4 cos^{2} (x) : - 5 cos^{2} (x) + 4

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

条件のようなグループ

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

類似した元を足す:

- cos^{2} (x) - 4 cos^{2} (x) = - 5 cos^{2} (x)

= - 5 cos^{2} (x) + 4

= - 5 cos^{2} (x) + 4

= \frac{- 5 cos ^{2} ( x ) + 4}{cos ^{2} ( x )}

= \frac{- 5 cos ^{2} ( x ) + 4}{cos ^{2} ( x )}

拡張

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )} : \frac{4}{cos ^{2} ( x )} - 5

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )}

分数の規則を適用する:

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

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )} = \frac{4}{cos ^{2} ( x )} - \frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{4}{cos ^{2} ( x )} - \frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

キャンセル

\frac{5 cos ^{2} ( x )}{cos ^{2} ( x )} : 5

\frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

共通因数を約分する:

cos^{2} (x)

= 5

= \frac{4}{cos ^{2} ( x )} - 5

= - 5 + \frac{4}{cos ^{2} ( x )}

= \frac{4}{cos ^{2} ( x )} - 5

両辺を同じ形式にできることを証明した

\Rightarrow 真

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