a) Cho  $T=\cos x.\cos 2x.\cos 4x. \cos 8x. \cos 16x$
    Chứng minh:    $\sin 32 x=32.T.\sin x                                                           (1)$
b) Cho  $A=(\cos \frac{a}{2} +\cos \frac{b}{2} )(\cos \frac{a}{4}+\cos \frac{b}{4})(\cos \frac{a}{8}+\cos \frac{b}{8})(\cos \frac{a}{16}+\cos \frac{b}{16})$
    Chứng minh:  $\cos a - \cos b =16A.(\cos \frac{a}{16}-\cos \frac{b}{16})$
Đẳng thức $(1)$ hiển nhiên khi $x=k.2\pi$
Nếu $x\neq  k2\pi$ thì $\sin 2x, \sin 4x, \sin 8x, \sin 16 x$ đều khác $0$.
Ta có:   $\sin 2x= 2\sin x. \cos x$
              $\sin 4x =2\sin 2x. \cos 2x$
              $......................$
              $\sin 32 x= 2\sin 16 x .\cos 16 x$
Nhân vế với vế, ta được:
              $\sin 2x. \sin 4x... \sin 32 x  =  32 \sin x. \cos x. \sin 2x. \cos 2x...\sin 16x. \cos 16x$
                                                              $=  32T. \sin x. \sin 2x ... \sin 16x$
Đơn giản  $\sin 2x \neq  0, ... , \sin 16x \neq  0$. Ta có $\sin 32x=32 T \sin x          (1)$ đpcm.
b) Áp dụng công thức biến đổi tổng thành tích, ta có:
$A=16\cos \frac{a+b}{4}\cos \frac{a-b}{4} \cos \frac{a+b}{8}\cos \frac{a-b}{8}\cos \frac{a+b}{16}\cos \frac{a-b}{16}\cos \frac
{a+b}{32} \cos \frac{a-b}{32} $
Đặt:    $\begin{cases}x=\frac{a+b}{32}  \\ y= \frac{a-b}{32} \end{cases} \Rightarrow  x+y=\frac{a}{16}, x-y=\frac{b}{16}   $
và biểu thức $A=16\cos 8x. \cos 4x. \cos 2x. \cos x.\cos 8y. \cos 4y.\cos 2y.\cos y$
Theo câu a) :$\sin 16x=16\sin x.\cos 8x.\cos 4x.\cos 2x.\cos x$
                       $\sin 16y=16\sin y.\cos 8y.\cos 4y.\cos 2y.\cos y$
             hay   $\sin 16x. \sin 16y=16\sin x.\sin y.A                                       (1)$
Mặt khác      $\sin 16x.\sin 16y=\frac{1}{2}[\cos 16(x-y)-\cos (x+y)]=\frac{1}{2}(\cos b -\cos a)  $
                      $\sin x. \sin y=\frac{1}{2}[\sin (x-y)-\sin(x+y)]=\frac{1}{2} (\cos \frac{b}{16}-\cos \frac{a}{16}  )  $
Thay vào $(1)$ ta có:   $\cos a -\cos b =16(\cos \frac{a}{16}-\cos \frac{b}{16}  ).A$  (đpcm)

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