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Question Number 144936 by mathdanisur last updated on 30/Jun/21

Answered by mindispower last updated on 30/Jun/21

$${sec}^4\left(x\right)+{csc}^4\left(x\right)⩾\frac{2}{{cos}^2\left(x\right){sin}^2\left(x\right)}=\frac{8}{{\left(sin\left(2x\right)\right)}^2}$$$$⩾8$$$${sec}^6\left(y\right)+{csc}^6\left(y\right)⩾\frac{2}{{cos}^3\left(y\right){sin}^3\left(y\right)}=\frac{16}{{sin}^3\left(2y\right)}⩾16$$$${sec}^8\left(z\right)+{csc}^8\left(z\right)⩾\frac{2}{(cos\left(z\right)sin{\left(z\right)}^4}⩾32$$$$\iff ⩾{(8.16.32)}^{\frac{1}{3}}=16$$$$=ifonlyifsin\left(2x\right)=sin\left(2y\right)=sin\left(2z\right)=1$$$$\Rightarrow x=y=z=\frac{\pi }{4}$$

Commented by mathdanisur last updated on 30/Jun/21

$$alotcoolSerthanks$$

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