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人気のある三角関数 >

証明する cos^4(t)-sin^4(t)=1-2sin^2(t)

解

証明する

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

解

真

解答ステップ

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

左側を操作する

cos^{4} (t) - sin^{4} (t)

因数

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

cos^{4} (t) - sin^{4} (t)

cos^{4} (t) - sin^{4} (t)

を書き換え

(cos^{2} (t))^{2} - (sin^{2} (t))^{2}

cos^{4} (t) - sin^{4} (t)

指数の規則を適用する:

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

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

= cos^{4} (t) - (sin^{2} (t))^{2}

指数の規則を適用する:

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

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

= (cos^{2} (t))^{2} - (sin^{2} (t))^{2}

= (cos^{2} (t))^{2} - (sin^{2} (t))^{2}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

(cos^{2} (t))^{2} - (sin^{2} (t))^{2} = (cos^{2} (t) + sin^{2} (t)) (cos^{2} (t) - sin^{2} (t))

= (cos^{2} (t) + sin^{2} (t)) (cos^{2} (t) - sin^{2} (t))

因数

cos^{2} (t) - sin^{2} (t) : (cos (t) + sin (t)) (cos (t) - sin (t))

cos^{2} (t) - sin^{2} (t)

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (t) - sin^{2} (t) = (cos (t) + sin (t)) (cos (t) - sin (t))

= (cos (t) + sin (t)) (cos (t) - sin (t))

= (cos^{2} (t) + sin^{2} (t)) (cos (t) + sin (t)) (cos (t) - sin (t))

= (cos (t) + sin (t)) (cos (t) - sin (t)) (cos^{2} (t) + sin^{2} (t))

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

= (cos (t) + sin (t)) (cos (t) - sin (t)) \cdot 1

簡素化

= (cos (t) + sin (t)) (cos (t) - sin (t))

拡張

(cos (t) + sin (t)) (cos (t) - sin (t)) : cos^{2} (t) - sin^{2} (t)

(cos (t) + sin (t)) (cos (t) - sin (t))

2乗の差の公式を適用する:

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

a = cos (t), b = sin (t)

= cos^{2} (t) - sin^{2} (t)

= cos^{2} (t) - sin^{2} (t)

2倍角の公式を使用:

cos^{2} (t) - sin^{2} (t) = cos (2 t)

= cos (2 t)

2倍角の公式を使用:

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

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

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

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{tan ( x ) + tan ( y )}{cot ( x ) + cot ( y )} = tan (x) tan (y)

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p ro v e 1 = sec^{2} (2 x) - tan^{2} (2 x)