Proof: sin (x) + cos (x) = √2sin (x + π/4)

Hello:

sin (x) + cos (x) = √2 (1/√2 sin (x) + 1/√2 cos (x))

but : 1/√2 = cos (π/4) = sin (π/4)

sin (x) + cos (x) = √2 (cos (π/4) sin (x) + sin (π/4) cos (x)) = √2 sin (x + π/4)

because : cos (π/4) sin (x) + sin (π/4) cos (x) = sin (x + π/4) by identity:

sin (a+b) = sina cosb + cosa sinb