Now use the trig identity:
So we now have:
Take the negative at the front and cancel with the negative inside the second sin function and multiply the top and the bottom by
So we are left with:
Evaluating the limit by direct substitution:
That is the derivative of
For a proof that:
Very much like Andrews I.'s answer. The details are different.
and the limits:
#cos(x+Deltax)#in the numerator only. (Its would be OK to expand it in the denominator, but it is not necessary.)
Regroup to use the fundamental trigonometric limits.
Evaluate the limit using the fundamental trig limits and continuity of cosine.