\Final answer:
The critical points of sinx and cosx occur at x = (pi/4)*(1 + 2n), where n is an integer.
Step-by-step explanation:
The critical points of sinx and cosx occur when their derivatives are equal to zero. The derivative of sinx is cosx, and the derivative of cosx is -sinx. Setting cosx = -sinx, we have:
sinx - cosx = 0
sinx = cosx
Using trigonometric identities, we know that sinx = cos(pi/4 - x). So:
cos(pi/4 - x) = cosx
Therefore, pi/4 - x = x + 2pi*n, where n is an integer. Solving for x gives:
x = (pi/4)*(1 + 2n)
So the critical points of both sinx and cosx occur at x = (pi/4)*(1 + 2n), where n is an integer.