Question 1087518
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You are given that {{{sin(x)}}} + {{{1/sin(x)}}} = 2.    (1)

Obviously, (1) implies that sin(x) is positive. 


Let y = {{{sqrt(sin(x))}}}.   Then (1) becomes

{{{y^2}}} + {{{1/y^2}}} = 2,

which implies {{{y^2}}} - 2 + {{{1/y^2}}} = 0 = {{{(y-1/y)^2}}} = 0.


Hence, {{{y}}} = {{{1/y}}} and then y = 1.


Thus sin(x) = 1 and cos(x) = 0.


Now {{{(sin(x))^(-7)}}} + {{{(cos(x))^7}}} = 1.
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Solved.