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受欢迎的三角函数 >

证明 cot^2(x)=csc^2(x)(1-sin^2(x))

解答

证明

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

解答

真

求解步骤

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

调整左侧

cot^{2} (x)

用 sin, cos 表示

cot^{2} (x)

使用基本三角恒等式:

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

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

化简

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

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

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

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

使用三角恒等式改写

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

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

使用基本三角恒等式:

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

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

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

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

证明 sin(x)+cot(x)(cos(x))=csc(x)

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

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})