#P={x:sinx-cosx=sqrt2 cosx} and Q={x:sinx+cosx=sqrt2 sinx} #
For the relation
#sinx-cosx=sqrt2 cosx#
#=>1/sqrt2cosx-1/sqrt2sinx=-cosx#
#=>cos(pi/4)cosx-sin(pi/4)sinx=-cosx#
#=>cos(x+pi/4)=cos(pi-x)#
#=>x+pi/4=2npi+pi-x#
#=>x=npi+(3pi)/8" where " nin ZZ#
So #P={x:x=npi+(3pi)/8" where " nin ZZ}#
For the relation
#sinx+cosx=sqrt2 sinx#
#=>1/sqrt2cosx+1/sqrt2sinx=sinx#
#=>cos(pi/4)cosx+cos(pi/4)sinx=cos(pi/2-x)#
#=>cos(x-pi/4)=cos(pi/2-x)#
#=>x-pi/4=2npi+(pi/2-x) " where " n in ZZ#
#=>2x=2npi+(3pi)/4 " where " n in ZZ#
#=>x=npi+(3pi)/8" where " nin ZZ#
So #Q={x:x=npi+(3pi)/8" where " nin ZZ}#
Hence we have #P=Q#