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Question Number 211961 by Nadirhashim last updated on 25/Sep/24

$$\mathit{i}\mathit{f}{7}^{{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{x}}+7^{{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x}}=8\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{x}$$

Answered by efronzo1 last updated on 25/Sep/24

$$7^{\sin {}^2\mathrm{x}}=\mathrm{a}\Rightarrow 7^{\cos {}^2\mathrm{x}}=7^{1-\sin {}^2\mathrm{x}}=\frac{7}{\mathrm{a}}$$$$\Rightarrow \mathrm{a}+\frac{7}{\mathrm{a}}=8$$$$\Rightarrow {\mathrm{a}}^2-8\mathrm{a}+7=0$$$$\Rightarrow \mathrm{a}=1\mathrm{or}\mathrm{a}=7$$$$\{\begin{array}{l}{7}^{\sin {}^2\mathrm{x}}=1\Rightarrow \sin {}^2\mathrm{x}=0\\ 7^{\sin {}^2\mathrm{x}}=7\Rightarrow \sin {}^2\mathrm{x}=1\end{array}$$

Answered by BaliramKumar last updated on 26/Sep/24

$$x=0,\frac{\pi }{2}$$

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