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受欢迎的三角函数 >

证明 (cos(x))/(tan(x))=csc(x)-sin(x)

解答

证明

\frac{cos ( x )}{tan ( x )} = csc (x) - sin (x)

解答

真

求解步骤

\frac{cos ( x )}{tan ( x )} = csc (x) - sin (x)

调整左侧

\frac{cos ( x )}{tan ( x )}

用 sin, cos 表示

\frac{cos ( x )}{tan ( x )}

使用基本三角恒等式:

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

= \frac{cos ( x )}{\frac{s i n ( x )}{c o s ( x )}}

化简

\frac{cos ( x )}{\frac{s i n ( x )}{c o s ( x )}} : \frac{cos ^{2} ( x )}{sin ( x )}

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

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

cos (x) cos (x)

使用指数法则:

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

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

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

调整右侧

csc (x) - sin (x)

用 sin, cos 表示

csc (x) - sin (x)

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

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

化简

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

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

将项转换为分式:

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

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

因为分母相等，所以合并分式:

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

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

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

1 - sin (x) sin (x)

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

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

= 1 - sin^{2} (x)

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

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

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

使用三角恒等式改写

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

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

p ro v e sec^{2} (θ) csc^{2} (θ) = sec^{2} (θ) + csc^{2} (θ)

证明 (1-cos^2(x))sec^2(x)=tan^2(x)

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

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p ro v e sin (θ) cot (θ) sec (θ) = 1

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

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