# Please could somebody explain how the expression involving &#x03B8;<!-- θ --> that

Please could somebody explain how the expression involving $\theta$ that
$\frac{1+\mathrm{sin}\theta }{5+3\mathrm{tan}\theta -4\mathrm{cos}\theta }$
approximates to for small values of $\theta$ is $1-2\theta +4{\theta }^{2}$?
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Cahokiavv
Just compute the first terms of the Taylor series of $\frac{1+\mathrm{sin}\theta }{5+3\mathrm{tan}\theta -4\mathrm{cos}\theta }$, and you will get $1-2\theta +4{\theta }^{2}$.
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misurrosne
Approximate the numerator and denominator up to ${\theta }^{2}$ terms and get
$\frac{1+\theta +o\left({\theta }^{2}\right)}{1+3\theta +2{\theta }^{2}+o\left({\theta }^{2}\right)}=\left(1+\theta \right)\left(1-3\theta +7{\theta }^{2}\right)+o\left({\theta }^{2}\right)=1-2\theta +4{\theta }^{2}+o\left({\theta }^{2}\right).$