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受欢迎的三角函数 >

证明 sec(x)(1-sin^2(x))=cos(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

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

化简

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

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

分式相乘:

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

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

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

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

乘以:

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

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

去除括号:

(a) = a

= 1 - sin^{2} (x)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

约分:

cos (x)

= cos (x)

= cos (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 (1-cos^2(θ))(1+cot^2(θ))=1

p ro v e (1 - cos^{2} (θ)) (1 + cot^{2} (θ)) = 1

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p ro v e cot (x) sin (x) = cos (x)

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p ro v e sin (2 x) + cos (2 x) - 1 = 2 sin (x) (cos (x) - sin (x))

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

证明 cot(x)(1-cos^2(x))= 1/2 sin(2x)

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