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受欢迎的三角函数 >

证明 cot(x)-tan(x)=sec(x)(csc(x)-2sin(x))

解答

证明

cot (x) - tan (x) = sec (x) (csc (x) - 2 sin (x))

解答

真

求解步骤

cot (x) - tan (x) = sec (x) (csc (x) - 2 sin (x))

调整左侧

cot (x) - tan (x)

用 sin, cos 表示

cot (x) - tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

= sin (x) cos (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin (x) cos (x)

对于

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

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

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

使用三角恒等式改写

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

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

同类项相加:

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

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

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

调整右侧

sec (x) (csc (x) - 2 sin (x))

用 sin, cos 表示

(csc (x) - 2 sin (x)) sec (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

分式相乘:

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

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

1 \cdot (\frac{1}{sin ( x )} - 2 sin (x)) = \frac{1}{sin ( x )} - 2 sin (x)

1 \cdot (\frac{1}{sin ( x )} - 2 sin (x))

乘以:

1 \cdot (\frac{1}{sin ( x )} - 2 sin (x)) = (\frac{1}{sin ( x )} - 2 sin (x))

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

去除括号:

(a) = a

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

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

化简

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

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

将项转换为分式:

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

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

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 ( x ) sin ( x )}{sin ( x )}

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

1 - 2 sin (x) sin (x)

2 sin (x) sin (x) = 2 sin^{2} (x)

2 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)

= 2 sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 2 sin^{2} (x)

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

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

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

使用分式法则:

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

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

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

= \frac{1 - 2 sin ^{2} ( x )}{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+cot^2(-θ)=csc^2(θ)

p ro v e 1 + cot^{2} (- θ) = csc^{2} (θ)

证明 cos(x)(1+tan^2(x))=sec(x)

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

证明 (sec^2(x)-1)csc(x)=sin(x)sec^2(x)

p ro v e (sec^{2} (x) - 1) csc (x) = sin (x) sec^{2} (x)

证明 (1+csc(x))/(sec(x))-cot(x)=cos(x)

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

证明 1/(1+cos(x))+1/(1-cos(x))=2csc^2(x)

p ro v e \frac{1}{1 + cos ( x )} + \frac{1}{1 - cos ( x )} = 2 csc^{2} (x)