Double angle formulae for sin ⁡ 2 θ and cos ⁡ 2 θ in tangent form?

Question

Double angle formulae for   sin  ⁡  2  θ and   cos  ⁡  2  θ in tangent form?

Jakobe Yang · Accepted Answer

Using   tan  ⁡  (  θ  )  =            sin      ⁡      (      θ      )              cos      ⁡      (      θ      )                  2      tan      ⁡      (      θ      )              1      +              tan        2            ⁡      (      θ      )        =            2      tan      ⁡      (      θ      )        1    ⋅      1          1      +              tan        2            ⁡      (      θ      )        =            2      sin      ⁡      (      θ      )              cos      ⁡      (      θ      )        ⋅      1          1      +              tan        2            ⁡      (      θ      )      Now  1  +      tan    2    ⁡  (  θ  )  =  1  +                    sin        2            ⁡      (      θ      )                      cos        2            ⁡      (      θ      )        =                    cos        2            ⁡      (      θ      )      +              sin        2            ⁡      (      θ      )                      cos        2            ⁡      (      θ      )        =      1                  cos        2            ⁡      (      θ      )      In the first equation, changing this value            2      tan      ⁡      (      θ      )              1      +              tan        2            ⁡      (      θ      )        =            2      sin      ⁡      (      θ      )              cos      ⁡      (      θ      )        ⋅      1          1      +              tan        2            ⁡      (      θ      )        =            2      sin      ⁡      (      θ      )              cos      ⁡      (      θ      )        ⋅                    cos        2            ⁡      (      θ      )        1    =  2  sin  ⁡  (  θ  )  cos  ⁡  (  θ  )  =  sin  ⁡  (  2  θ  )Go through a similar process for   cos  ⁡  (  2  θ  )  .

Payton Salazar · Accepted Answer

sin  ⁡  2  θ  =  2  sin  ⁡  θ  cos  ⁡  θ  =            2      sin      ⁡      θ      cos      ⁡      θ                      cos        2            ⁡      θ      +              sin        2            ⁡      θ        =            2      tan      ⁡      θ              1      +              tan        2            ⁡      θ        cos  ⁡  2  θ  =      cos    2    ⁡  θ  −      sin    2    ⁡  θ  =                    cos        2            ⁡      θ      −              sin        2            ⁡      θ                      cos        2            ⁡      θ      +              sin        2            ⁡      θ        =            1      −              tan        2            ⁡      θ              1      +              tan        2            ⁡      θ

Double angle formulae for sin ⁡<!-- ⁡

Answered question

Answer & Explanation

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