Find the period of cos ⁡ x − sin ⁡ xShow that f had period of 2 πHow do you find the period of a function that has several trigonometric functions like this one?I tried converting them all to sine: f ( x ) = 1 − sin 2 ⁡ x − sin ⁡ xBut what do I do next?

Question

Find the period of   cos  ⁡  x  −  sin  ⁡  xShow that f had period of   2  πHow do you find the period of a function that has several trigonometric functions like this one?I tried converting them all to sine:  f  (  x  )  =      1    −          sin      2        ⁡    x    −  sin  ⁡  xBut what do I do next?

exhumatql · Accepted Answer

You can rewrite:  f  (  x  )  =      cos    ⁡    x    −  sin  ⁡  x  =      sin    ⁡          (              π        2            −      x      )        −  sin  ⁡  xand then use:  sin  ⁡  a  −  sin  ⁡  b  =  2  cos  ⁡      (                  a        +        b            2        )    sin  ⁡      (                  a        −        b            2        )  to find that f(x) can also be written as:  f  (  x  )  =      2    sin  ⁡      (          π      4        −    x    )

Xiomara Poole · Accepted Answer

Rewrite   cos  ⁡  x  −  sin  ⁡  x to   −      2    sin  ⁡  (  x  −  π      /    4  ) and it is obvious that period is   2  πAnother way : T&amp;gt;0 then  cos  ⁡  (  x  +  T  )  −  sin  ⁡  (  x  +  T  )  =  cos  ⁡  x  −  sin  ⁡  x  →  sin  ⁡  x  −  sin  ⁡  (  x  +  T  )  =  cos  ⁡  x  −  cos  ⁡  (  x  +  T  )  →              2  sin  ⁡  (            −      T        2    )  cos  ⁡  (            2      x      +      T        2    )  =  −  2  sin  ⁡  (            2      x      +      T        2    )  sin  ⁡  (            −      T        2    )  →  sin  ⁡  (            −      T        2    )  (  cos  ⁡  (            2      x      +      T        2    )  +  sin  ⁡  (            2      x      +      T        2    )  )  =  0

Find the period of cos ⁡ x − sin ⁡

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