Question

How do you sketch the curve `y=cos^2x-sin^2x` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

Answer 1

Note that we can use the trigonometric identity:

`cos^2x-sin^2x = cos 2x`

so we know that the function has a maximum for `x = kpi` and a minimum for `x= pi/2+kpi` with `k in ZZ`.

The inflection points are coincident with the intercepts and are in `x= pi/4+kpi`. The function is concave down in `(-pi/4+kpi, pi/4+kpi)` with `k` even, and concave up with `k` odd.

Finally, the function is continuous for every `x in RR` and has no limit for `x->+-oo` and no asymptotes.

graph{cos(2x) [-2.5, 2.5, -1.25, 1.25]}