证明 cot^3(x)=cot(x)(csc^2(x)-1)

解答

证明

cot^{3} (x) = cot (x) (csc^{2} (x) - 1)

真

求解步骤

cot^{3} (x) = cot (x) (csc^{2} (x) - 1)

调整右侧

cot (x) (csc^{2} (x) - 1)

使用三角恒等式改写

cot (x) (csc^{2} (x) - 1)

使用毕达哥拉斯恒等式:

csc^{2} (x) = 1 + cot^{2} (x)

csc^{2} (x) - 1 = cot^{2} (x)

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

化简

cot (x) cot^{2} (x) : cot^{3} (x)

cot (x) cot^{2} (x)

使用指数法则:

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

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

= cot^{1 + 2} (x)

数字相加:

1 + 2 = 3

= cot^{3} (x)

= cot^{3} (x)

= cot^{3} (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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

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

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

p ro v e tan (2 A) - 2 tan (2 A) sin^{2} (A) = sin (2 A)

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