For what values does $\sin(x+h) = \cos x$? For what and how many values of h such that $0 \leq h \leq 100$ are the graphs $a(x) = \sin(x+h)$ and $b(x) = \cos x$  identical?
I am not sure where to start. Hints only please.
 A: Hint:
$$\sin(h+x)=\cos(x)\implies \sin(h)\cos(x)+\cos(h)\sin(x)=\cos x$$
Now, for what value of h does $\cos(h)=0, \sin(h)=1?$
Here's a unit circle for reference:

Also note that $\sin(2\pi+x) , \cos(2\pi+x)=\sin(x) , \cos(x)$
Therefore, when you find $h$, add multiples of $2\pi$ to obtain all the possible values of $h$.
A: Try the formula
$$
\sin(x+h) = \sin(x)\cos(h) + \cos(x)\sin(h)
$$
and plug in suitable values of $x$.
A: both functions are $2\pi$ periodic. so if you know how many times $a$ and $b$ are identical for $h$ in one period $0 \le x 2\pi,$ then you can figure out how many periods there are in $100$ multiply by the number of coincidences in one period.
now you are back to figuring out the number of times $\sin(x+h)$ and $\cos(x)$ are identical for $0 \le h \le 2\pi.$ how many translation to the left of the $y = y=\sin(x)$ match the graph $y = \cos(x)$
hint: (i) draw the  graphs $y = y=\sin(x)$ and $y = \cos(x)$ for $0 \le x \le 2\pi$
      (ii) keeping the $x$ and $y$ axes fixed translate the $\sin x$ graph left and see where it coincides with that of the fixes $\cos x$ 
A: $$\sin(x+h)=\cos x=\sin\left(\frac\pi2-x\right)\iff$$
$$\iff\begin{cases}x+h=\frac\pi2-x\iff x=\frac\pi4-\frac h2\\{}\\
x+h=\pi-\left(\frac\pi2-x\right)\iff h=\frac\pi2\end{cases}\;\;\;+2k\pi\;,\;\;k\in\Bbb Z$$
