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受欢迎的三角函数 >

证明 sin(x)+cot(x)(cos(x))=csc(x)

解答

证明

sin (x) + cot (x) (cos (x)) = csc (x)

解答

真

求解步骤

sin (x) + cot (x) cos (x) = csc (x)

调整左侧

sin (x) + cot (x) cos (x)

用 sin, cos 表示

sin (x) + cos (x) cot (x)

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

= \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 )}

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

将项转换为分式:

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

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

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

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

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{\frac{1}{c s c ( x )}}

化简

\frac{1}{\frac{1}{c s c ( x )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

使用法则

\frac{a}{1} = a

= csc (x)

csc (x)

csc (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/(csc(x)-1)=(sin(x))/1

p ro v e \frac{1}{csc ( x ) - 1} = \frac{sin ( x )}{1}

证明 sec(θ)cos(θ)csc(θ)=cot(θ)

p ro v e sec (θ) cos (θ) csc (θ) = cot (θ)

证明 csc^2(x)(1-cos^2(x))=tan(420)

p ro v e csc^{2} (x) (1 - cos^{2} (x)) = tan (42 0^{\circ})

证明 cos(θ+30)-sin(θ+60)=-sin(θ)

p ro v e cos (θ + 3 0^{\circ}) - sin (θ + 6 0^{\circ}) = - sin (θ)

证明 tan(a)*cot(a)=sin^2(a)+cos^2(a)

p ro v e tan (a) \cdot cot (a) = sin^{2} (a) + cos^{2} (a)