Prove that? sinx + cosx= √2cos(π/4 -x)

Question

14582 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-08T13:56:56

See below.

Identity:

#color(red)bb(cos(A-B)=cosAcosB+sinAsinB)#

#RHS#

#sqrt(2)cos(pi/4-x)=sqrt(2)[cos(pi/4)cos(x)+sin(pi/4)sin(x)]#

#cos(pi/4)=sqrt(2)/2#

#sin(pi/4)=sqrt(2)/2#

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=sqrt(2)[sqrt(2)/2cos(x)+sqrt(2)/2sin(x)]#

Factoring:

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=(sqrt(2)sqrt(2))/2[cos(x)+sin(x)]#

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=cos(x)+sin(x)#

#RHS-=LHS#