Xét tính đúng sai của các mệnh đề sau:

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

a) Đúng. cosa + cos2a + cos3a + cos4a = (cosa + cos3a) + (cos2a + cos4a)

= 2cos2acosa + 2cos3acosa = 2(cos2a + cos3a)cosa = \(4\cos \frac{{5a}}{2} \cdot \cos \frac{a}{2} \cdot \cos a\).

b) Đúng. 1 + sina + cos3a – (cosa + sin2a + cos2a)

= (1 − cos2a) + (cos3a – cosa) + (sina – sin2a)

= 2sin²a – 2sin2asina + (sina – sin2a)

= 2sina(sina – sin2a) + (sina – sin2a) = (sina – sin2a)(2sina +1)

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

c) Đúng. cosa + cosb + cosc + cos(a + b + c)

= (cosa + cosb) + [cosc + cos(a + b +c )]

\( = 2\cos \frac{{a + b}}{2} \cdot \cos \frac{{a - b}}{2} + 2\cos \frac{{c + a + b + c}}{2} \cdot \cos \frac{{c + a + b - c}}{2}\)

\( = 2\cos \frac{{a + b}}{2} \cdot \cos \frac{{a - b}}{2} + 2\cos \frac{{a + b + 2c}}{2} \cdot \cos \frac{{a + b}}{2}\)

\( = 2\cos \frac{{a + b}}{2} \cdot \left( {\cos \frac{{a - b}}{2} + \cos \frac{{a + b + 2c}}{2}} \right) = 2\cos \frac{{a + b}}{2} \cdot 2\cos \frac{{b + c}}{2} \cdot \cos \frac{{c + a}}{2}\)

\( = 4\cos \frac{{a + b}}{2} \cdot \cos \frac{{b + c}}{2} \cdot \cos \frac{{c + a}}{2}\).

d) Sai. sin x + sin(x + a) + sin(x + 2a)

= [sin(x + 2a) + sinx] + sin(x + a) = \(2\sin \frac{{x + 2a + x}}{2} \cdot \cos \frac{{x + 2a - x}}{2} + \sin (x + a)\)

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

= (2cosa + 1)sin(x + a).