Expand the left side using the angle sum identity for cosine:
[tex]\cos(\theta+\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi[/tex]
This reduces to [tex]\cos\theta[/tex] if the sine product is 0 and [tex]\cos\phi=1[/tex]. This happens when [tex]\phi=2n\pi[/tex] where [tex]n[/tex] is any integer; that is, any even multiple of [tex]\pi[/tex].
So technically the claim is false; it's true for infinitely many values of [tex]\phi[/tex]. But it is certainly true for [tex]\phi=2\pi[/tex] rad.
