证明 (cos(3x)-cos(x))=-2sin(2x)sin(x)

解答

证明

(cos (3 x) - cos (x)) = - 2 sin (2 x) sin (x)

真

求解步骤

cos (3 x) - cos (x) = - 2 sin (2 x) sin (x)

调整左侧

cos (3 x) - cos (x)

使用三角恒等式改写

cos (3 x) - cos (x)

使用和差化积恒等式:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{3 x + x}{2}) sin (\frac{3 x - x}{2})

化简

- 2 sin (\frac{3 x + x}{2}) sin (\frac{3 x - x}{2}) : - 2 sin (2 x) sin (x)

- 2 sin (\frac{3 x + x}{2}) sin (\frac{3 x - x}{2})

\frac{3 x + x}{2} = 2 x

\frac{3 x + x}{2}

p ro v e sec (\frac{π}{2} - y) = csc (y)

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

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

p ro v e cos (x) \cdot csc (x) \cdot tan (x) = 1

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