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受欢迎的三角函数 >

证明 cot^2(x)=((cos^2(x)))/(1-cos^2(x))

解答

证明

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

解答

真

求解步骤

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

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

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

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

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

证明 sin((4pi)/3)=sqrt(3)cos((2pi)/3)

p ro v e sin (\frac{4 π}{3}) = 3 cos (\frac{2 π}{3})

证明 sin^2(x)+sin^2(θ)=1

p ro v e sin^{2} (x) + sin^{2} (θ) = 1

证明 (cot(x)-sec(x))/(csc(x))=1

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

证明 1+tan^2(x)+cot(x)=sec^2(x)+cos(x)

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