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Question Number 185036 by Mastermind last updated on 16/Jan/23

$$\mathrm{Find}\mathrm{the}\mathrm{series}\mathrm{for}\mathrm{cosx}.\mathrm{Hence},$$$$\mathrm{deduce}\mathrm{series}{\sin }^2\mathrm{x}\mathrm{and}\mathrm{show}\mathrm{that},$$$$\mathrm{if}\mathrm{x}\mathrm{is}\mathrm{small},\frac{{\sin }^2\mathrm{x}-{\mathrm{x}}^2\mathrm{cosx}}{{\mathrm{x}}^4}=\frac{1}{6}+\frac{{\mathrm{x}}^2}{360}$$$$\mathrm{approximately}.$$$$$$$$$$$$\mathrm{Help}!$$

Answered by aba last updated on 16/Jan/23

$$\cos \left(\mathrm{x}\right)=\underset{\mathrm{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{\mathrm{n}}{\mathrm{x}}^{2\mathrm{n}}}{\left(2\mathrm{n}\right)!}$$$${\sin }^2\left(\mathrm{x}\right)=\frac{1-\cos \left(2\mathrm{x}\right)}{2}=\frac{1}{2}[1-\underset{\mathrm{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{\mathrm{n}}{\left(2\mathrm{x}\right)}^{2\mathrm{n}}}{\left(2\mathrm{n}\right)!}]$$

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