# Assume that angle x is small, using small angle approximation, sin(x)=x; cos(x)=1-(x^2)/2; and tan(x)=x. I am able to justify the first two using Maclaurin's Theorem but not the last one. How do we justify the last one?

Justify the small angle approximation for tangent
Assume that angle x is small, using small angle approximation,
$\mathrm{sin}\left(x\right)=x$
$\mathrm{cos}\left(x\right)=1-\frac{{x}^{2}}{2}$;
and $\mathrm{tan}\left(x\right)=x$.
I am able to justify the first two using Maclaurin's Theorem but not the last one. How do we justify the last one?
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Kendrick Jacobs
Explanation:
For small x, $\frac{1}{\mathrm{cos}x}\approx \frac{1}{1-\frac{{x}^{2}}{2}}\approx 1+\frac{{x}^{2}}{2}$.

glyperezrl
Step 1
Fairly simple actually: $\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}\approx \frac{x}{1-\frac{{x}^{2}}{2}}=x\left(1+\frac{{x}^{2}}{2}+\frac{{x}^{4}}{4}+\dots \right)\approx x\left(1+\mathcal{O}\left({x}^{2}\right)\right)$
Step 2
Where we used known approximations and the geometric series.