Xét tính đúng sai của các mệnh đề dưới đây:

Đáp án: a) Sai. b) Sai. c) Sai. d) Sai.

a) Sai. sina − sin2a + sin3a = (sina + sin3a) – sin2a

\( = 2\sin \frac{{a + 3a}}{2} \cdot \cos \frac{{a - 3a}}{2} - \sin 2a = 2\sin 2a\cos a - \sin 2a\)

\( = \sin 2a \cdot \left( {2\cos a - 1} \right)\).

b) Sai. sinx + 2sin3x + sin5x = (sinx + sin5x) + 2sin3x

\( = 2\sin \frac{{x + 5x}}{2} \cdot \cos \frac{{x - 5x}}{2} + 2\sin 3x = 2\sin 3x \cdot \cos 2x + 2\sin 3x = 2\sin 3x \cdot \left( {\cos 2x + 1} \right)\)

\( = 2\sin 3x \cdot 2{\cos ^2}x = 4\sin 3x \cdot {\cos ^2}x\).

c) Sai. 2cosx – cos3x – cos5x = 2cosx – (cos3x + cos5x)

\( = 2\cos x - 2\cos \frac{{3x + 5x}}{2} \cdot \cos \frac{{3x - 5x}}{2} = 2\cos x - 2\cos 4x \cdot \cos x = 2\cos x\left( {1 - \cos 4x} \right)\)

\( = 2\cos x \cdot 2{\sin ^2}2x = 4\cos x \cdot {\sin ^2}2x\).

d) Sai. \[\frac{{\cos 5x - \cos x}}{{\sin 4x + \sin 2x}} = \frac{{ - 2\sin \frac{{5x + x}}{2} \cdot \sin \frac{{5x - x}}{2}}}{{2\sin \frac{{4x + 2x}}{2} \cdot \cos \frac{{4x - 2x}}{2}}}\]

\[ = \frac{{ - 2\sin 3x \cdot \sin 2x}}{{2\sin 3x \cdot \cos x}} = - \frac{{2\sin x\cos x}}{{\cos x}} = - 2\sin x\].