Solve.1. sin ⁡ x cos 3 ⁡ x − cos ⁡ x sin 3 ⁡ x = 1 / 4 sin ⁡ 4 x2. cos ⁡ x + cos ⁡ 4 x + cos ⁡ 7 x sin ⁡ x + sin ⁡ 4 x + sin ⁡ 7 x = cot ⁡ 4 x

Question

Solve.1.  sin  ⁡  x      cos          3        ⁡  x  −  cos  ⁡  x      sin          3        ⁡  x  =  1      /    4  sin  ⁡  4  x2.            cos      ⁡      x      +      cos      ⁡      4      x      +      cos      ⁡      7      x              sin      ⁡      x      +      sin      ⁡      4      x      +      sin      ⁡      7      x        =  cot  ⁡  4  x

Monastro3n · Accepted Answer

1) Take out common factors:  sin  ⁡  x  cos  ⁡  x  [      cos          2        ⁡  x  −      sin          2        ⁡  x  ]  =  (  1      /    4  )  sin  ⁡  (  4  x  )double angle identity:  sin  ⁡  (  2  x  )  =  2  sin  ⁡  (  x  )  cos  ⁡  (  x  )So,            sin      ⁡      (      2      x      )        2    [  cos  ⁡  (  2  x  )  ]  =            sin      ⁡      (      4      x      )        4              sin      ⁡      (      2      x      )      cos      ⁡      (      2      x      )        2    =            sin      ⁡      (      4      x      )        4  We already know that:  sin  ⁡  (  2  u  )  =  sin  ⁡  (  u  )  cos  ⁡  (  u  )if we let u = 2x:  sin  ⁡  [  2  (  2  x  )  ]  =  sin  ⁡  (  2  x  )  cos  ⁡  (  2  x  )  sin  ⁡  (  4  x  )  =  sin  ⁡  (  2  x  )  cos  ⁡  (  2  x  )            sin      ⁡      (      4      x      )        4    =            sin      ⁡      (      4      x      )        4  2) we know that   cos  ⁡  x  +  cos  ⁡  7  x  =  2  cos  ⁡  (  7  x  +  x  )      /    2  ∗  c  o  s  (  7  x  −  x  )      /    2  =  2  cos  ⁡  4  x  cos  ⁡  3  xand   sin  ⁡  x  +  sin  ⁡  7  x  =  2  sin  ⁡  (  7  x  +  x  )      /    2  ∗  cos  ⁡  (  7  x  −  x  )      /    2  =  2  sin  ⁡  4  x  cos  ⁡  3  x            cos      ⁡      x      +      cos      ⁡      4      x      +      cos      ⁡      7      x              sin      ⁡      x      +      sin      ⁡      4      x      +      sin      ⁡      7      x        =            2      cos      ⁡      4      x      ∗      cos      ⁡      3      x      +      cos      ⁡      4      x              2      sin      ⁡      4      x      ∗      cos      ⁡      3      x      +      sin      ⁡      4      x        =            cos      ⁡      4      x      (      2      cos      ⁡      3      x      +      1      )              sin      ⁡      4      x      (      2      cos      ⁡      3      x      +      1      )        =            cos      ⁡      4      x              sin      ⁡      4      x        =  cot  ⁡  x

Solve. 1. sin x cos^3 x-cosx sin^3 x=1/4sin^4x (cos x+cos 4x+cos 7x)/(sin x+sin 4x+sin 7x)=cot 4x

Answered question

Answer & Explanation

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