trigonometry - Rewriting an expression in the form of $A sin(x + C)$

The problem asks to rewrite

$$\sin(x) - \cos(x)$$ in the form of $A\sin(x + C)$, using the reduction formula.

The answer is supposed to be $\sqrt{2}\sin(x - \pi/4)$, or $\sqrt{2}\sin(x - 45)$ using degrees.

But from what the book is doing, I don't know what C is supposed to be or how to get it.

Answer

This is an interesting question, because you wouldn't necessarily thing that the sum of two sine waves would give another sine wave and not some other wavy thing. The key to solving this problem is using two identities. The first is the complementary angle one: $\cos x = \sin (\pi/2-x)$ using radians. The second (which you may be calling the reduction formula) is the sum-to-product formula $\sin a - \sin b = 2 \cos ((a+b)/2) \sin ((a-b)/2)$. Applying these,

$$\sin x - \cos x = \sin x - \sin (\pi/2 -x) = 2\cos(\pi/4) \sin (x-\pi/4) = \sqrt{2} \sin (x-\pi/4)$$

which is close to the answer you gave.

The "$C$" is essentially the phase shift of the new sign function that occurs because you are making a new function. Alternatively, you could take Simon's hint (which may or may not be more obvious) by expanding $\sqrt{2} \sin (x+\pi/4)$ and working backwards.