The general solution of tan ⁡ ( x ) = tan ⁡ ( 2 x − π / 2 )

Question

The general solution of   tan  ⁡  (  x  )  =  tan  ⁡  (  2  x  −      π        /    2  )

Savanna Smith · Accepted Answer

tan  ⁡      (    2    x    −          π      2        )    =  −  tan  ⁡      (          π      2        −    2    x    )    =  −      1          tan      ⁡              2        x              tan  ⁡      x    =  −      1          tan      ⁡              2        x              →  tan  ⁡      x    tan  ⁡      2    x    =  −  1  tan  ⁡      x              2      tan      ⁡              x                    1      −              tan        2            ⁡              x              =  −  1  →  2      tan    2    ⁡      x    =  −  1  +      tan    2    ⁡      x    →      tan    2    ⁡      x    =  −  1  !

Alberto Calhoun · Accepted Answer

If you set   t  =  tan  ⁡  x and apply the duplication formula for the tangent, you have  tan  ⁡            (        2  x  −      π    2              )        =  −      1          tan      ⁡      2      x        =                    t        2            −      1              2      t        ,so the equation becomes                    t        2            −      1              2      t        =  t    ⟺        t    2    −  1  =  2      t    2      ⟺        t    2    =  −  1  ,which has no real solution

The general solution of tan(x)=tan(2x−pi/2)

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