HMMT 二月 2010 · 冲刺赛 · 第 26 题

英文解析

26. [ 15 ] Express the following in closed form, as a function of x :
    2 2 2 2
    2 2 2 2010 2 2009 2 2
    sin ( x )+sin (2 x ) cos ( x )+sin (4 x ) cos (2 x ) cos ( x )+ · · · +sin (2 x ) cos (2 x ) · · · cos (2 x ) cos ( x ) .
    2 2011
    sin (2 x )
    Answer: 1 − Note that
    20112
    4 sin ( x )
    2 2 2 2 2010 2 2009 2 2
    sin ( x ) + sin (2 x ) cos ( x ) + · · · + sin (2 x ) cos (2 x ) · · · cos (2 x ) cos ( x )
    2 2 2 2 2010 2 2009 2 = (1 − cos ( x )) + (1 − cos (2 x )) cos ( x ) + · · · + (1 − cos (2 x )) cos (2 x ) · · · cos ( x ) ,
    2 2 2 2 2010 2 2 2 2010
    which telescopes to 1 − cos ( x ) cos (2 x ) cos (4 x ) · · · cos (2 x ). To evaluate cos ( x ) cos (2 x ) · · · cos (2 x ),
    multiply and divide by sin ( x ). We then get 2
    2 2 2 20102
    sin ( x ) cos ( x ) cos (2 x ) · · · cos (2 x )
    1 − .
    sin ( x )2
    sin (2 y )2
    Using the double-angle formula for sin, we get that sin ( y ) cos ( y ) = . Applying this 2011 times 22
    makes the above expression 4
    20112
    sin (2 x )
    1 − ,
    20112
    4 sin ( x )
    which is in closed form.