When $\theta$ is small, ${\theta }^4$ and higher powers are extremely small, so

Interestingly, the diagram below suggests that

1. Find the sum of the infinite geometric series

   $$1+\frac{{\theta }^2}{2}+\frac{{\theta }^4}{4}+\frac{{\theta }^6}{8}\dots$$

   and hence show that, for small values of $\theta$, this sum is approximately equal to $\sec \theta .$

2. Given that

   $$1-\frac{x^2}{2}\le \cos x\le \frac{1}{1+\frac{x^2}{2}}$$

   for all small values of $x,$ show that the error between $\cos x$ and the small angle approximation $1-\frac{x^2}{2}$ is smaller than

   $$\frac{x^4}{2(x^2+2)}$$

3. Hence, show that the error given by the small angle approximation for $x=0.1$ is smaller than $2.5\times {10}^{-5}$

For (a), use the formula

for geometric series. You should see the small angle approximation for $\cos \theta$ in your answer.

For (b), look at the graph to see that the error between

is smaller than the error between