Solve the equation 1 / cos ⁡ x = cos ⁡ x + sin ⁡ x

Question

Solve the equation   1      /    cos  ⁡  x  =  cos  ⁡  x  +  sin  ⁡  x

Ethen Valentine · Accepted Answer

1  =      cos    2    ⁡  x  +  cos  ⁡  x  sin  ⁡  x    ⟺    cos  ⁡  x  sin  ⁡  x  =      sin    2    ⁡  x    ⟺    sin  ⁡  x  (  cos  ⁡  x  −  sin  ⁡  x  )  =  0If   sin  ⁡  x  =  0  ,  x  =  n  πElse   cos  ⁡  x  −  sin  ⁡  x  =  0    ⟺    tan  ⁡  x  =  1    ⟹    x  =  m  π  +            π      4      where m,n are integers

vittorecostao1 · Accepted Answer

1          cos      ⁡      (      x      )        −  cos  ⁡  (  x  )  =  sin  ⁡  (  x  )  .Squaring and setting       cos    2    ⁡  (  x  )  =  t      1    t    −  2  +  t  =  1  −  t  .The solutions of this second degree equation   2      t    2    −  3  t  +  1  =  0 are t=1 and   t  =      1    2  , corresponding to  cos  ⁡  (  x  )  =  ±  1  →  x  =  k  π  ,    cos  ⁡  (  x  )  =  ±      1          2        →      π    4    +            k      π        2    .Anyway, squaring the equation introduced extra solutions. After checking compatibility, we are left with  x  =  k  π  ,    x  =      π    4    +  k  π  .

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