Question

Упростите выражения
sin(pi/4-a)-cosa

Answer 1

$sin( frac{pi}{4} -a)-cos(a)=sin( frac{pi}{4} )cos(a)-cos( frac{pi}{4} )sin(a)-cos(a)=$
$= frac{ sqrt{2} }{2} cos(a)-frac{ sqrt{2} }{2}sin(a)-cos(a)= frac{ sqrt{2}-1 }{2} cos(a)-frac{ sqrt{2} }{2}sin(a)$

$sin( frac{pi}{4} -a)-cos(a)= sin( frac{pi}{4} -a)-sin( frac{pi}{2} +a)=$
$=2sin( frac{frac{pi}{4} -a-a}{2} )cos( frac{frac{pi}{4} -a+a}{2} )=2*cos frac{pi}{8}*sin( frac{pi}{8}-a ) =$
$=2*cos (frac{pi}{8})*sin( frac{pi}{8}-a ) = =2* sqrt{ frac{1+cos( frac{pi}{4} )}{2} } *sin( frac{pi}{8}-a ) =$
$=2* sqrt{ frac{1+ frac{ sqrt{2} }{2} }{2} } *sin( frac{pi}{8}-a ) =2* frac{ sqrt{2+ sqrt{2} } }{2} *sin( frac{pi}{8}-a ) =$
$=sqrt{2+ sqrt{2} } *sin( frac{pi}{8}-a ) =-sqrt{2+ sqrt{2} } *sin( a-frac{pi}{8} )$