Mathematics

Problem

Let

g(x)=\sin x

for all real

x

. a. (i) Show that

\sin(x+h)-\sin x=2\cos\!\left(x+\dfrac{h}{2}\right)\sin\!\left(\dfrac{h}{2}\right)

. b. (ii) Using the result of part (i), evaluate

\displaystyle\lim_{h\to 0}\frac{\sin(x+h)-\sin x}{h}

for an arbitrary real number

x

. Give your answer in its simplest trigonometric form. c. (iii) State the derivative of

g

at an arbitrary point

x