ordinary differential equations - Sine/Cosine Subtraction Formulas

The differential equation $\frac{d^2 y}{dt^2}+ \omega^2 y=0$ has the general solution $y = A\cos (\omega t)+ B\sin (\omega t)$. Also given are the initial values: $y(a) = A, y'(a) = B$.

I tried:

$$y(a) = A\cos(\omega a) + B\sin(\omega a )= A$$

$$y'(a) = −\omega A\sin(\omega a) + \omega B\cos(\omega a)= B$$

And then substituted $A$ and $B$ into the general solution:

$$y = (A\cos(\omega a) + B\sin(\omega a))\cosωt + \omega (−A\sin(\omega a) + B\cos(\omega a))\sin(\omega t)$$

From here I can't get it to work. I want to simplify and express y with the subtraction formulas.
The answer is $y = A\cos(\omega (t−a)) + \frac{B}{\omega}\sin(\omega (t−a))$

I know how the formulas work, But I can't figure out how $\frac{B}{\omega}$ got there.
Please help, I'm stuck.