The value of $sin 73° + cos 137°$ is :

$sin13°$

sin 73° + cos 137°

= sin ( 60° + 13° ) + cos ( 180° - 43°)

{ cos is negative in 2nd quadrant }

= sin ( 60° + 13° ) - cos ( 43°) 

= sin ( 60° + 13° ) - cos ( 30° + 13°)

{ using , sin  ( A + B ) = sinA . cosB + cosA .sinB 

& cos( A + B ) = cosA . cosB - sinA . sinB }

= sin60°.cos13° + cos60°.sin13° - cos30° . cos13° + sin30°.sin13°

= \(\frac{√3 }{2}\).cos13° + \(\frac{1 }{2}\).sin13° - \(\frac{√3 }{2}\). cos13° + \(\frac{1 }{2}\)°.sin13°

= sin 13°