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If $\sin x+\operatorname{cosec} x=2$, then what is the value of $\sin ^{4} x+\operatorname{cosec}^{4} x ?$
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Given $\sin x+\operatorname{cosec} x=2$
$\begin{aligned} \text { Consider } \sin ^{4} x+\operatorname{cosec}^{4} x=\left(\sin ^{2} x+\operatorname{cosec}^{2} x\right)^{2} \\ &-2\left(\sin ^{2} x \cos ^{2} x\right) \\=& \left.\left[(\sin x+\operatorname{cosec} x)^{2}-2\right)\right]^{2}-2 \\=(4-2)^{2}-2=2 & \end{aligned}$
$\begin{aligned} \text { Consider } \sin ^{4} x+\operatorname{cosec}^{4} x=\left(\sin ^{2} x+\operatorname{cosec}^{2} x\right)^{2} \\ &-2\left(\sin ^{2} x \cos ^{2} x\right) \\=& \left.\left[(\sin x+\operatorname{cosec} x)^{2}-2\right)\right]^{2}-2 \\=(4-2)^{2}-2=2 & \end{aligned}$
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