Mathematics

Problem

Establish and analyse an identity involving a sum of sine and cosine. a. (a) Show that

\cos x = \sin\left(\dfrac{\pi}{2}-x\right)

. b. (b) Using the result of part (a), prove that

\sin x + \cos x = \sqrt2\,\sin\left(x+\dfrac{\pi}{4}\right)

for all real

x

. c. (c) Hence determine the maximum value of

\sin x + \cos x

and the set of all

x \in [0,2\pi)

for which this maximum occurs.