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受欢迎的三角函数 >

证明 1/(sin(x))-sin(x)=cot(x)cos(x)

解答

证明

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

解答

真

求解步骤

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

调整右侧

cot (x) cos (x)

用 sin, cos 表示

cos (x) cot (x)

使用基本三角恒等式:

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

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

化简

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 )}

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

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

乘开

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

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

使用分式法则:

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

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

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

消掉

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

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

约分:

sin (x)

= sin (x)

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

= \frac{1}{sin ( 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 sin(3x)=3sin(x)-4(sin(x))^3

p ro v e sin (3 x) = 3 sin (x) - 4 (sin (x))^{3}

证明 (sin(θ)-1)/(cos(θ))=tan(θ)-sec(θ)

p ro v e \frac{sin ( θ ) - 1}{cos ( θ )} = tan (θ) - sec (θ)

证明 cot(x)*sec(x)=csc(x)

p ro v e cot (x) \cdot sec (x) = csc (x)

证明 2-csc^2(θ)=1-cot^2(θ)

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

证明 1-(cos^2(x))/(1-sin(x))=-sin(x)

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