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Prove the following:
\[\log \ \sec \ x = \Big [ \dfrac{x^2}{2}+\dfrac{x^4}{12}+\dfrac{x^6}{45}... \Big]\]
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written 4.6 years ago by
teamques10
★ 70k
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$log(1+\sin x)=\sin x-\dfrac{\sin^{2}x}{2}+\dfrac{\sin^{3}x}{3}-...$ $=\left ( x-\dfrac{x^{3}}{3!}+\dfrac{x^{5}}{5!}-... \right )-\dfrac{1}{2}\left ( x-\dfrac{x^{3}}{3!} +...\right )^{2}+\dfrac{1}{3}\left ( x-\dfrac{x^{3}}{3!}+... \right )^{3}-...$ $=x-\dfrac{1}{2}x^{2}-\dfrac{x^{3}}{3!}+\dfrac{1}{3}x^{3}+...$ $=x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{6}+....$
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