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受欢迎的三角函数 >

证明 cot^2(θ)(1-cos^2(θ))=cos^2(θ)

解答

证明

cot^{2} (θ) (1 - cos^{2} (θ)) = cos^{2} (θ)

解答

真

求解步骤

cot^{2} (θ) (1 - cos^{2} (θ)) = cos^{2} (θ)

调整左侧

cot^{2} (θ) (1 - cos^{2} (θ))

用 sin, cos 表示

(1 - cos^{2} (θ)) cot^{2} (θ)

使用基本三角恒等式:

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

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

化简

(1 - cos^{2} (θ)) (\frac{cos ( θ )}{sin ( θ )})^{2} : \frac{cos ^{2} ( θ ) ( 1 - cos ^{2} ( θ ) )}{sin ^{2} ( θ )}

(1 - cos^{2} (θ)) (\frac{cos ( θ )}{sin ( θ )})^{2}

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

(\frac{cos ( θ )}{sin ( θ )})^{2}

使用指数法则:

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

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

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

分式相乘:

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

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

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

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

使用三角恒等式改写

\frac{( 1 - cos ^{2} ( θ ) ) cos ^{2} ( θ )}{sin ^{2} ( θ )}

使用毕达哥拉斯恒等式:

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

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

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

约分:

sin^{2} (θ)

= cos^{2} (θ)

= cos^{2} (θ)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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