证明 sin(5x)-sin(3x)=2cos(4x)sin(x)

解答

证明

sin (5 x) - sin (3 x) = 2 cos (4 x) sin (x)

真

求解步骤

sin (5 x) - sin (3 x) = 2 cos (4 x) sin (x)

调整左侧

sin (5 x) - sin (3 x)

使用三角恒等式改写

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

化简

2 sin (\frac{5 x - 3 x}{2}) cos (\frac{5 x + 3 x}{2}) : 2 sin (x) cos (4 x)

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

\frac{5 x - 3 x}{2} = x

\frac{5 x - 3 x}{2}

p ro v e sin (4 0^{\circ}) = 2 sin (2 0^{\circ}) cos (2 0^{\circ})

p ro v e sin (θ) - cos^{2} (θ) sin (θ) = sin^{3} (θ)

p ro v e sec^{2} (x) + csc^{2} (x) = (sec (x) csc (x))^{2}

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

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