I have been trying to simplify this expression: cos ⁡ ( θ ) − 1 cos ⁡ ( θ ) + i sin ⁡ ( θ )

Question

I have been trying to simplify this expression:  cos  ⁡  (  θ  )  −      1          cos      ⁡      (      θ      )      +      i      sin      ⁡      (      θ      )

Kaydence Washington · Accepted Answer

c  o  s  (  θ  )  −      1          cos      ⁡      (      θ      )      +      i      sin      ⁡      (      θ      )        =  c  o  s  (  θ  )  −      1          e              i        θ              =  c  o  s  (  θ  )  −      e          −      i      θ        =      1    2        e          i      θ        +      1    2        e          −      i      θ        −      e          −      i      θ        =  =      1    2        e          i      θ        −      1    2        e          −      i      θ        =  i            (            1          2      i            e          i      θ        −      1          2      i            e          −      i      θ                  )        =  i  sin  ⁡  (  θ  )

Gaaljh · Accepted Answer

HINT:      e          −      i      θ        =  cos  ⁡  (  −  θ  )  +  i  sin  ⁡  (  −  θ  )  =  cos  ⁡  θ  −  i  sin  ⁡  θ    ⟹    cos  ⁡  θ  −  i  sin  ⁡  θ  =            1              cos        ⁡        θ        +        i        sin        ⁡        θ            which is also evident from   (  cos  ⁡  θ  +  i  sin  ⁡  θ  )  (  cos  ⁡  θ  −  i  sin  ⁡  θ  )  =  ⋯  =  1

I have been trying to simplify this expression: cos ⁡ ( θ<!-- θ --

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