zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

证明 cot(2θ)=(csc(θ)-2sin(θ))/(2cos(θ))

解答

证明

cot (2 θ) = \frac{csc ( θ ) - 2 sin ( θ )}{2 cos ( θ )}

解答

真

求解步骤

cot (2 θ) = \frac{csc ( θ ) - 2 sin ( θ )}{2 cos ( θ )}

调整右侧

\frac{csc ( θ ) - 2 sin ( θ )}{2 cos ( θ )}

用 sin, cos 表示

\frac{csc ( θ ) - 2 sin ( θ )}{2 cos ( θ )}

使用基本三角恒等式:

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

= \frac{\frac{1}{s i n ( θ )} - 2 sin ( θ )}{2 cos ( θ )}

化简

\frac{\frac{1}{s i n ( θ )} - 2 sin ( θ )}{2 cos ( θ )} : \frac{1 - 2 sin ^{2} ( θ )}{2 sin ( θ ) cos ( θ )}

\frac{\frac{1}{s i n ( θ )} - 2 sin ( θ )}{2 cos ( θ )}

化简

\frac{1}{sin ( θ )} - 2 sin (θ) : \frac{1 - 2 sin ^{2} ( θ )}{sin ( θ )}

\frac{1}{sin ( θ )} - 2 sin (θ)

将项转换为分式:

2 sin (θ) = \frac{2 sin ( θ ) sin ( θ )}{sin ( θ )}

= \frac{1}{sin ( θ )} - \frac{2 sin ( θ ) sin ( θ )}{sin ( θ )}

因为分母相等，所以合并分式:

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

= \frac{1 - 2 sin ( θ ) sin ( θ )}{sin ( θ )}

1 - 2 sin (θ) sin (θ) = 1 - 2 sin^{2} (θ)

1 - 2 sin (θ) sin (θ)

2 sin (θ) sin (θ) = 2 sin^{2} (θ)

2 sin (θ) sin (θ)

使用指数法则:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= 2 sin^{1 + 1} (θ)

数字相加:

1 + 1 = 2

= 2 sin^{2} (θ)

= 1 - 2 sin^{2} (θ)

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

= \frac{\frac{1 - 2 s i n ^{2} ( θ )}{s i n ( θ )}}{2 cos ( θ )}

使用分式法则:

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

= \frac{1 - 2 sin ^{2} ( θ )}{sin ( θ ) \cdot 2 cos ( θ )}

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

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

使用三角恒等式改写

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

使用倍角公式:

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

= \frac{cos ( 2 θ )}{2 cos ( θ ) sin ( θ )}

使用倍角公式:

2 sin (x) cos (x) = sin (2 x)

= \frac{cos ( 2 θ )}{sin ( 2 θ )}

= \frac{cos ( 2 θ )}{sin ( 2 θ )}

使用基本三角恒等式:

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

= cot (2 θ)

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 sin^2(θ)sec^2(θ)csc^2(θ)=sec^2(θ)

p ro v e sin^{2} (θ) sec^{2} (θ) csc^{2} (θ) = sec^{2} (θ)

证明 (csc^2(x))/(cot(x))=sec(x)csc(x)

p ro v e \frac{csc ^{2} ( x )}{cot ( x )} = sec (x) csc (x)

证明 (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A)

p ro v e \frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )} = 2 sec (A)

证明 cot(x/2)=(sin(x))/(1-cos(x))

p ro v e cot (\frac{x}{2}) = \frac{sin ( x )}{1 - cos ( x )}

证明-tan(y)+sec(y)=(cos(y))/(1+sin(y))

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