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受欢迎的三角函数 >

证明 3csc(x)-cot(x)=(3-cos(x))/(sin(x))

解答

证明

3 csc (x) - cot (x) = \frac{3 - cos ( x )}{sin ( x )}

解答

真

求解步骤

3 csc (x) - cot (x) = \frac{3 - cos ( x )}{sin ( x )}

调整左侧

3 csc (x) - cot (x)

用 sin, cos 表示

- cot (x) + 3 csc (x)

使用基本三角恒等式:

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

= - \frac{cos ( x )}{sin ( x )} + 3 csc (x)

使用基本三角恒等式:

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

= - \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{1}{sin ( x )}

化简

- \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{1}{sin ( x )} : \frac{- cos ( x ) + 3}{sin ( x )}

- \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{1}{sin ( x )}

3 \cdot \frac{1}{sin ( x )} = \frac{3}{sin ( x )}

3 \cdot \frac{1}{sin ( x )}

分式相乘:

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

= \frac{1 \cdot 3}{sin ( x )}

数字相乘:

1 \cdot 3 = 3

= \frac{3}{sin ( x )}

= - \frac{cos ( x )}{sin ( x )} + \frac{3}{sin ( x )}

使用法则

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

= \frac{- cos ( x ) + 3}{sin ( x )}

= \frac{- cos ( x ) + 3}{sin ( x )}

= \frac{3 - cos ( 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 1/(1-sin(θ))=sec^2(θ)+sec(θ)tan(θ)

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

证明 2sin(x)=tan(x)

p ro v e 2 sin (x) = tan (x)

证明 cot(x)-1=csc(x)(cos(x)-sin(x))

p ro v e cot (x) - 1 = csc (x) (cos (x) - sin (x))

证明-cos(x)=sin(x+pi/2)

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

证明 cos(4t)=1-8sin^2(t)cos^2(t)

p ro v e cos (4 t) = 1 - 8 sin^{2} (t) cos^{2} (t)