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受欢迎的三角函数 >

证明 (1-sin(x))/(sec(x))=(cos^3(x))/(1+sin(x))

解答

证明

\frac{1 - sin ( x )}{sec ( x )} = \frac{cos ^{3} ( x )}{1 + sin ( x )}

解答

真

求解步骤

\frac{1 - sin ( x )}{sec ( x )} = \frac{cos ^{3} ( x )}{1 + sin ( x )}

调整左侧

\frac{1 - sin ( x )}{sec ( x )}

使用三角恒等式改写

乘以

\frac{1 + sin ( x )}{1 + sin ( x )}

= \frac{( 1 + sin ( x ) ) ( 1 - sin ( x ) )}{( 1 + sin ( x ) ) sec ( x )}

乘开

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

(1 + sin (x)) (1 - sin (x))

使用平方差公式:

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

a = 1, b = sin (x)

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (x)

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

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

= \frac{cos ^{2} ( x )}{( 1 + sin ( x ) ) \frac{1}{c o s ( x )}}

化简

\frac{cos ^{2} ( x )}{( 1 + sin ( x ) ) \frac{1}{c o s ( x )}} : \frac{cos ^{3} ( x )}{1 + sin ( x )}

\frac{cos ^{2} ( x )}{( 1 + sin ( x ) ) \frac{1}{c o s ( x )}}

乘

(1 + sin (x)) \frac{1}{cos ( x )} : \frac{1 + sin ( x )}{cos ( x )}

(1 + sin (x)) \frac{1}{cos ( x )}

分式相乘:

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

= \frac{1 \cdot ( 1 + sin ( x ) )}{cos ( x )}

1 \cdot (1 + sin (x)) = 1 + sin (x)

1 \cdot (1 + sin (x))

乘以:

1 \cdot (1 + sin (x)) = (1 + sin (x))

= (1 + sin (x))

去除括号:

(a) = a

= 1 + sin (x)

= \frac{1 + sin ( x )}{cos ( x )}

= \frac{cos ^{2} ( x )}{\frac{1 + s i n ( x )}{c o s ( x )}}

使用分式法则:

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

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

cos^{2} (x) cos (x) = cos^{3} (x)

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

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{2 + 1} (x)

数字相加:

2 + 1 = 3

= cos^{3} (x)

= \frac{cos ^{3} ( x )}{1 + sin ( x )}

= \frac{cos ^{3} ( x )}{1 + sin ( x )}

= \frac{cos ^{3} ( x )}{1 + sin ( x )}

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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