Question 1152875
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<pre>

Let x be the greater of the two given acute angles, and let y be the smaller.


Then, first, the equality  tan(x)*tan(y) = 1  implies that

    x + y = 90°.         (1)


Second, 

    tan(x-y) = {{{(tan(x)-tan(y))/(1 + tan(x)*tan(y))}}} = 

         substitute given values for the numerator and denominator to get

              = {{{((2*sqrt(3)))/(1+1)}}} = {{{((2*sqrt(3)))/2}}} = {{{sqrt(3)}}},

which implies  

    x - y = 60°.           (2)


From equations (1) and (2), by adding, you get

    2x = 90° + 60° = 150°;   hence,  x = {{{150^o/2}}} = 75°.


Finally, substituting this value of x into (1), you get  y = 15°.


So, under the given conditions,  x = 75°  and  y = 15°.


In particular,  x = 5y.
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Solved.