Question

Упростите выражения sin(pi/4-a)-cosa

Answer 1

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

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