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

Jul 26, 2017

Note that we can use the trigonometric identity:

${\cos}^{2} x - {\sin}^{2} x = \cos 2 x$

so we know that the function has a maximum for $x = k \pi$ and a minimum for $x = \frac{\pi}{2} + k \pi$ with $k \in \mathbb{Z}$.

The inflection points are coincident with the intercepts and are in $x = \frac{\pi}{4} + k \pi$. The function is concave down in $\left(- \frac{\pi}{4} + k \pi , \frac{\pi}{4} + k \pi\right)$ with $k$ even, and concave up with $k$ odd.

Finally, the function is continuous for every $x \in \mathbb{R}$ and has no limit for $x \to \pm \infty$ and no asymptotes.

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