The key to trigonometry is

The key to trigonometry is ${\cos }^2\theta =1-{\sin }^2\theta$.
Set $\sin \theta \approx \theta$ to find ${\cos }^2\theta \approx 1-{\theta }^2$. The square root is $\cos \theta \approx 1-\frac{1}{2}{\theta }^2$. Reason: Squaring gives ${\cos }^2\theta \approx$ ____ and the correction term ____ is very small near $\theta =0$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §2.4 The Derivative of the Sine and Cosine, #7