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受欢迎的三角函数 >

证明 1/(cos(x))-cos(x)=sin(x)tan(x)

解答

证明

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

解答

真

求解步骤

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

调整右侧

sin (x) tan (x)

用 sin, cos 表示

sin (x) tan (x)

使用基本三角恒等式:

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

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

化简

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

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

分式相乘:

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

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

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

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

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

使用三角恒等式改写

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

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

乘开

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

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

使用分式法则:

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

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

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

消掉

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

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

约分:

cos (x)

= cos (x)

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

= - cos (x) + \frac{1}{cos ( x )}

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

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

证明 sin(60)=2sin(30)cos(30)

p ro v e sin (6 0^{\circ}) = 2 sin (3 0^{\circ}) cos (3 0^{\circ})

证明 (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ)

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

证明 sec(2θ)=(sec^2(θ))/(1-tan^2(θ))

p ro v e sec (2 θ) = \frac{sec ^{2} ( θ )}{1 - tan ^{2} ( θ )}

证明 cos(2a)=cos^2(a)-sin^2(a)

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