Small angle approximations

Take a look at a graph of $y = \cos (x)$ and compare it with $y = 1 - \frac{x^{2}}{2}$ .

cos(x) and 1-x^2/2

When $x$ is close to $0,$ the curves are almost the same. This means that when $x$ is small, we can work with $1 - \frac{1}{x^{2}}$ instead of $\cos (x),$ which can sometimes make problems much simpler.

The situation is similar with $y = \sin (x)$ and $y = x$

sin(x) and x

and again with $y = \tan (x)$ and $y = x$

tan(x) and x

So, when $θ$ is small (i.e. close to $0$ ):

\begin{aligned} \cos θ & \approx 1 - \frac{θ^{2}}{2} \\ \sin θ & \approx θ \\ \tan θ & \approx θ \end{aligned}

These are called the small angle approximations, and they tend to work best when $θ$ is smaller than about $0.2$ radians, although this really depends on the level of accuracy required for your particular application.

It is important to note that the small angle approximations are only valid when $θ$ is measured in radians.

Given that $x$ is small and measured in radians, use the small angle approximation to find an expression which approximates

f (x) = \cosec x (\cos 2 x + \tan \frac{x}{2})

simplifying your answer.