Principal solution of given trigonometric equationFind the principal solution of the following equation: sin 2 ⁡ x + 2 tan 2 ⁡ x + 4 3 tan ⁡ x − sin ⁡ x + 11 12 = 0I tried it by converting both sin ⁡ x and tan ⁡ x into half angles ( of tan ⁡ x / 2) but it is resulting in ugly equation. Could someone suggest a smarter approach without using half angles?

Question

Principal solution of given trigonometric equationFind the principal solution of the following equation:      sin    2    ⁡  x  +  2      tan    2    ⁡  x  +      4          3        tan  ⁡  x  −  sin  ⁡  x  +      11    12    =  0I tried it by converting both   sin  ⁡  x and   tan  ⁡  x into half angles ( of   tan  ⁡  x      /    2) but it is resulting in ugly equation. Could someone suggest a smarter approach without using half angles?

Dahlia Pace · Accepted Answer

On some manipulation of the   L  H  S of the equation, it can be shown that it is the same as  (  s  i  n  (  x  )  −      1    2        )    2    +  2  (  t  a  n  (  x  )  +      1                  (            3      )            )    2  . And since squares of real numbers are always non-negative, for the sum of the 2 terms to be 0, both of them have to be simultaneously 0.And both of them are simultaneously zero at   x  =  2  n  π  +            5      π        6   for   n  ∈  ZThe principal solution is thus             5      π        6

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