Question 1185683
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According to the addition formula for cosine arguments, the left side is  cos(5x+3x) = cos(8x).


Therefore, the given equation is EQUIVALENT to


    cos(8x) = {{{sqrt(3)/2}}}


and has the solutions for 8x


    8x = {{{pi/6}}},  {{{pi/6 + 2pi}}},  {{{pi/6 + 4pi}}},  {{{pi/6 + 6pi}}}, . . . , {{{pi/6 + 14pi}}}  (8 values)

or

    8x = {{{11pi/6}}},  {{{11pi/6 + 2pi}}},  {{{11pi/6 + 4pi}}},  {{{11pi/6 + 6pi}}}, . . . , {{{11pi/6 + 14pi}}}  (another 8 values).


Dividing by 8, we get these 8 different solutions for x


    x = {{{pi/48}}},  {{{pi/48 + 2pi/8}}},  {{{pi/48 + 4pi/8}}},  {{{pi/48 + 6pi/8}}}, . . . , {{{pi/48 + 14pi/8}}}  (8 values)

or

    x = {{{11pi/48}}},  {{{11pi/48 + 2pi/8}}},  {{{11pi/48 + 4pi/8}}},  {{{11pi/48 + 6pi/8}}}, . . . , {{{11pi/48 + 14pi/8}}}  (another 8 values).


The listed 8 + 8 = 16 values for x represent the full set of solutions to given equation in the interval [0,2pi).


You can make obvios reducing of fractions. I left the fraction in this form in order for you can better see the structure of the solution.
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Solved, explained and completed.