Solve the identy. 1 + sin ⁡ 2 x cos ⁡ x + sin ⁡ x = cos ⁡ 2 x cos ⁡ x − sin ⁡ x

Question

Solve the identy.             1      +      sin      ⁡      2      x              cos      ⁡      x      +      sin      ⁡      x        =            cos      ⁡      2      x              cos      ⁡      x      −      sin      ⁡      x

Penelope Powers · Accepted Answer

left side:             cos      ⁡      2      x              cos      ⁡      x      −      sin      ⁡      x        =                    cos                  2                    ⁡      x      −              sin                  2                    ⁡      x              cos      ⁡      x      −      sin      ⁡      x        =            (      cos      ⁡      x      −      sin      ⁡      x      )      (      cos      ⁡      x      +      sin      ⁡      x      )              cos      ⁡      x      −      sin      ⁡      x        =  cos  ⁡  x  +  sin  ⁡  xright side:             1      +      sin      ⁡      2      x              cos      ⁡      x      +      sin      ⁡      x        =            (              cos                  2                    ⁡      x      +              sin                  2                    ⁡      x      )      +      2      sin      ⁡      x      cos      ⁡      x              cos      ⁡      x      +      sin      ⁡      x        (      cos          2        ⁡  x  +      sin          2        ⁡  x  =  1     a  n  d     sin  ⁡  2  x  =  2  sin  ⁡  x  cos  ⁡  x  )  =            (      cos      ⁡      x      +      sin      ⁡      x              )                  2                            cos      ⁡      x      +      sin      ⁡      x        =  cos  ⁡  x  +  sin  ⁡  x

Jimena Hatfield · Accepted Answer

consider the right hand side :i.e  cos  ⁡  2  x  =  [      cos          2        ⁡  x  −      sin          2        ⁡  x  ]      /    cos  ⁡  x  −  sin  ⁡  x            cos      ⁡      2      x              cos      ⁡      x      −      sin      ⁡      x        =  [      cos          2        ⁡  x  −      sin          2        ⁡  x  ]      /    cos  ⁡  x  −  sin  ⁡  x  =  (  cos  ⁡  x  −  sin  ⁡  x  )  (  cos  ⁡  x  +  sin  ⁡  x  )      /    (  cos  ⁡  x  −  sin  ⁡  x  )            a              2              −          b              2              =    (    a    +    b    )    (    a    −    b    )    =  cos  ⁡  x  +  sin  ⁡  x  =  (  cos  ⁡  x  +  sin  ⁡  x      )          2            /    (  cos  ⁡  x  +  sin  ⁡  x  )  =  [      cos          2        ⁡  x  +      sin          2        ⁡  x  +  2  sin  ⁡  x  cos  ⁡  x  ]      /    (  cos  ⁡  x  +  sin  ⁡  x  )  =  [  1  +  sin  ⁡  2  x  ]      /    (  cos  ⁡  x  +  sin  ⁡  x  )  =  L  H  SHence proved

Solve the identy. (1+sin 2x)/(cos x+sin x)=(cos 2x)/(cos x-sin x)

Answered question

Answer & Explanation

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