From … find the Taylor series up to

From ${\sec }^{2}x=1/[1-{\sin }^{2}x]$ find the Taylor series up to $x^2$. Check by squaring the secant series in Problem 37. Check by differentiating the tangent series in Problem 39.

This exercise builds on another exercise in its source book, which isn't shown here.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §10.5 Power Series, #38