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人気のある三角関数 >

証明する cos(4x)=1-8sin^2(x)+8sin^4(x)

解

証明する

cos (4 x) = 1 - 8 sin^{2} (x) + 8 sin^{4} (x)

解

真

解答ステップ

cos (4 x) = 1 - 8 sin^{2} (x) + 8 sin^{4} (x)

左側を操作する

cos (4 x)

三角関数の公式を使用して書き換える

cos (4 x)

= cos (2 \cdot 2 x)

2倍角の公式を使用:

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

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

2倍角の公式を使用:

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

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

簡素化

2 (1 - 2 sin^{2} (x))^{2} - 1 : 8 sin^{4} (x) - 8 sin^{2} (x) + 1

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

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

完全平方式を適用する:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

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

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

簡素化

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

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

数を乗じる:

2 \cdot 1 \cdot 2 = 4

= 4 sin^{2} (x)

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

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

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} b^{n}

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

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

指数の規則を適用する:

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

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

数を乗じる:

2 \cdot 2 = 4

= sin^{4} (x)

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

2^{2} = 4

= 4 sin^{4} (x)

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

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

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

拡張

2 (1 - 4 sin^{2} (x) + 4 sin^{4} (x)) : 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

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

括弧を分配する

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

マイナス・プラスの規則を適用する

+ (- a) = - a

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

簡素化

2 \cdot 1 - 2 \cdot 4 sin^{2} (x) + 2 \cdot 4 sin^{4} (x) : 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

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

数を乗じる:

2 \cdot 1 = 2

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

数を乗じる:

2 \cdot 4 = 8

= 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

= 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

= 2 - 8 sin^{2} (x) + 8 sin^{4} (x) - 1

簡素化

2 - 8 sin^{2} (x) + 8 sin^{4} (x) - 1 : 8 sin^{4} (x) - 8 sin^{2} (x) + 1

2 - 8 sin^{2} (x) + 8 sin^{4} (x) - 1

条件のようなグループ

= - 8 sin^{2} (x) + 8 sin^{4} (x) + 2 - 1

数を足す/引く:

2 - 1 = 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 1 - 8 sin^{2} (x) + 8 sin^{4} (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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p ro v e \frac{sec ( x )}{csc ( x )} = tan (x)

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p ro v e cot^{2} (θ) (1 + tan^{2} (θ)) = csc^{2} (θ)

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証明する cos^4(θ)-sin^4(θ)=2cos^2(θ)-1

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