how to find Taylor series of $\cos(x^2 +y^2)$

Question

Compute the degree ten Taylor polynomial of $\cos(x^2 +y^2)$ based at the origin.

Answer 1

We have that

$$\cos t = 1-\frac12 t^2+\frac1{24}t^4 +O(t^6)$$

then by $t=x^2+y^2$

$$\cos (x^2+y^2) = 1-\frac12 (x^2+y^2)^2+\frac1{24}(x^2+y^2)^4 +O((x^2+y^2)^6)$$