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人気のある三角関数 >

証明する (tan^2(x)+1)(cos^2(x)-1)=-tan^2(x)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

拡張

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

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

FOIL メソッドを適用する:

(a + b) (c + d) = a c + a d + b c + b d

a = - 1, b = cos^{2} (x), c = 1, d = tan^{2} (x)

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

マイナス・プラスの規則を適用する

+ (- a) = - a

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

簡素化

- 1 \cdot 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x) : - 1 - tan^{2} (x) + cos^{2} (x) + cos^{2} (x) tan^{2} (x)

- 1 \cdot 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x)

数を乗じる:

1 \cdot 1 = 1

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

乗算:

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

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

乗算:

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

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

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

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

簡素化

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

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

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

共通因数を約分する:

cos^{2} (x)

= sin^{2} (x)

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

類似した元を足す:

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

= - tan^{2} (x)

= - tan^{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

証明する csc(θ)cos(θ)=cot(θ)

p ro v e csc (θ) cos (θ) = cot (θ)

証明する sec(x)= 1/(cos(x))

p ro v e sec (x) = \frac{1}{cos ( x )}

証明する csc^2(x)-cot^2(x)=1

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

証明する cot(x)-tan(x)=2cot(2x)

p ro v e cot (x) - tan (x) = 2 cot (2 x)

証明する sin^4(x)-cos^4(x)=2sin^2(x)-1

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