sqr 2/2(sinx + cosx)= sin (x + pi/4)
√(2)/2*(sin(x) + cos(x)) = sin(x + π/4)
Expanding the rhs gives us,
sin(x + π/4) = sin(x)cos(π/4) + cos(x) sin(π/4)
using sin(π/4) = cos(π/4) = 1/√2.
sin(x + π/4) = sin(x)(1/√2) + cos(x)(1/√2)
sin(x + π/4) = (1/√2){sin(x) + cos(x)}
sin(x + π/4) = (√2/2){sin(x) + cos(x)} = lhs
i.e. the expression is an identity.