证明 sin(5x)-sin(x)=2cos(3x)sin(2x)

解答

证明

sin (5 x) - sin (x) = 2 cos (3 x) sin (2 x)

真

求解步骤

sin (5 x) - sin (x) = 2 cos (3 x) sin (2 x)

调整左侧

sin (5 x) - sin (x)

使用三角恒等式改写

sin (5 x) - sin (x)

使用和差化积恒等式:

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

= 2 sin (\frac{5 x - x}{2}) cos (\frac{5 x + x}{2})

化简

2 sin (\frac{5 x - x}{2}) cos (\frac{5 x + x}{2}) : 2 sin (2 x) cos (3 x)

2 sin (\frac{5 x - x}{2}) cos (\frac{5 x + x}{2})

\frac{5 x - x}{2} = 2 x

\frac{5 x - x}{2}

p ro v e tan (\frac{π}{3}) sin (\frac{2}{3} π) = 1 + sin (\frac{π}{6})

p ro v e csc (2 x) = \frac{2 3}{3}

p ro v e tan (x) = sin (x) - cos (x)

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

p ro v e tan (\frac{11}{12} π) = tan (- \frac{π}{12})