Prove that sin 4 ⁡ θ + cos 4 ⁡ θ = 3 + cos ⁡ 4 θ 4 By considering ( sin 2 ⁡ θ + cos 2 ⁡ θ ) 2 and ( sin 2 ⁡ θ + cos 2 ⁡ θ ) 3 prove that a) sin 4 ⁡ θ + cos 4 ⁡ θ = 3 + cos ⁡ 4 θ 4 , b) sin 6 ⁡ θ + cos 6 ⁡ θ = 5 + 3 cos ⁡ 4 θ 8

Question

Prove that       sin    4    ⁡  θ  +      cos    4    ⁡  θ  =            3      +      cos      ⁡      4      θ        4  By considering   (      sin    2    ⁡  θ  +      cos    2    ⁡  θ      )    2   and   (      sin    2    ⁡  θ  +      cos    2    ⁡  θ      )    3   prove that  a)       sin    4    ⁡  θ  +      cos    4    ⁡  θ  =            3      +      cos      ⁡      4      θ        4    ,    b)       sin    6    ⁡  θ  +      cos    6    ⁡  θ  =            5      +      3      cos      ⁡      4      θ        8

candelo6a · Accepted Answer

a) Since       sin    2    ⁡  (  θ  )  +      cos    2    ⁡  (  θ  )  =  1 we can write:   1  =      1    2    =  (      sin    2    ⁡  (  θ  )  +      cos    2    ⁡  (  θ  )      )    2    =      cos    4    ⁡  (  θ  )  +      sin    4    ⁡  (  θ  )  +  2      sin    2    ⁡  (  θ  )      cos    2    ⁡  (  θ  ). Hence       cos    4    ⁡  (  θ  )  +      sin    4    ⁡  (  θ  )  =  1  −  2      sin    2    ⁡  (  θ  )      cos    2    ⁡  (  θ  )  =  1  −      1    2        sin    2    ⁡  (  2  θ  )  =  1  −      1    2              1      −      cos      ⁡      (      4      θ      )        2  , the claim follows. b) is similar, even there you just have to remember basic goniometric ids (in particular bisection and dupliaction).

opepayflarpws · Accepted Answer

Hint: For a  L  .  H  .  S  =      sin    4    ⁡  A  +      cos    4    ⁡  A  =  (      sin    2    ⁡  A  +      cos    2    ⁡  A      )    2    −  2      sin    2    ⁡  A  .      cos    2    ⁡  A  =  1  −      1    4    (  2  sin  ⁡  A  .  cos  ⁡  A      )    2    =  1  −      1    4        sin    2    ⁡  2  A  =            4      −              sin        2            ⁡      2      A        4    =            4      −      1      +      cos      ⁡      4      A        4    =            3      +      cos      ⁡      4      A        4    =  R  .  H  .  SProved.Just consider that   θ  =  AFollow similar steps above for b.

Prove that sin 4 </msup> ⁡ θ + cos

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