O valor de: $$\tan^{10}x - 5\tan^{8}x \sec^{2}x + 10\tan^{6}x \sec^{4}x - 10\tan^{4}x \sec^6x + 5\tan^{2}x \sec^8x -\sec ^{10}x$$para todo $x \in [0 , \pi /2[$ é:


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ITA IIIT 29/04/2022 01:46
$-$ Veja que: \begin{matrix} (\tan^2{x} - \sec^2{x})^5 &=& \tan^{10}{x} &-& 5\tan^{8}{x}\sec^{2}{x} &+& 10\tan^{6}{x}\sec^{4}{x} &-& 10\tan^{4}{x}\sec^{6}{x} &-& 5\tan^{2}{x}\sec^{8}{x} &+& \sec^{10}{x} \end{matrix} \begin{matrix} (\tan^2{x} - \sec^2{x})^5 &=& [\tan^2{x} -(1+\tan^2{x})]^5 &=& (-1)^5 &=& -1 \end{matrix} \begin{matrix} Letra \ (D) \end{matrix}$\color{orangered}{Obs:}$ $\sec^2{x} = 1 + \tan^2{x}$
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