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受欢迎的三角函数 >

证明 cos(t)cot(t)=(1-sin^2(t))/(sin(t))

解答

证明

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

解答

真

求解步骤

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

调整左侧

cos (t) cot (t)

用 sin, cos 表示

cos (t) cot (t)

使用基本三角恒等式:

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

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

化简

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

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

分式相乘:

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

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

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

cos (t) cos (t)

使用指数法则:

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

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

= cos^{1 + 1} (t)

数字相加:

1 + 1 = 2

= cos^{2} (t)

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

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

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

使用三角恒等式改写

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

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} ( t )}{sin ( t )}

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

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^2(2u)-sin^2(2u)=cos(4u)

p ro v e cos^{2} (2 u) - sin^{2} (2 u) = cos (4 u)

证明 tan(u)cot(u)-cos^2(u)=sin^2(u)

p ro v e tan (u) cot (u) - cos^{2} (u) = sin^{2} (u)

证明 cos(2θ)=1-2sin^2(θ)

p ro v e cos (2 θ) = 1 - 2 sin^{2} (θ)

证明 1/(csc(x)-sin(x))=tan(x)sec(x)

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

证明 sin(3x)=(sin(x))(3-4sin^2(x))

p ro v e sin (3 x) = (sin (x)) (3 - 4 sin^{2} (x))