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Algebraic Proof that

if f(x) = a sin(bx + c) and g(x) = a cos(bx + c),

then the SUM of f and g is equivalent to

h(x) = asqrt(2)cos(bx - (p/4 - c)).

(1) Work with f + g:

f(x) + g(x) = a(sin bx cos c + cos bx sin c + cos bx cos c - sin bx sin c)

(2) Work with h:

h(x) = asqrt(2)[cos bx cos(p/4 - c) + sin bx sin(p/4 - c)]

= asqrt(2)[cos bx(cos p/4 cos c + sin p/4 sin c) + sin bx(sin p/4 cos c - cos p/4 sin c)]

= asqrt(2)[sqrt(2)/2 cos bx cos c + sqrt(2)/2 cos bx sin c + sqrt(2)/2 sin bx cos c - sqrt(2)/2 sin bx sin c)]

= a(cos bx cos c + cos bx sin c + sin bx cos c - sin bx sin c)

= a(sin bx cos c + cos bx sin c + cos bx cos c - sin bx sin c) = f(x) + g(x) above. QED.