How to prove these identities are true?1) sin ⁡ ( 2 θ ) = 2 sin ⁡ ( θ ) cos ⁡ ( θ )2) cos ⁡ ( 2 θ ) = cos 2 ⁡ θ − sin 2 ⁡ θ = 1 − 2 sin 2 ⁡ θ = 2 cos 2 ⁡ θ − 13) tan ⁡ 2 θ = 2 tan ⁡ θ 1 − tan 2 ⁡ θ

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How to prove these identities are true?1)  sin  ⁡  (  2  θ  )  =  2  sin  ⁡  (  θ  )  cos  ⁡  (  θ  )2)  cos  ⁡  (  2  θ  )  =      cos    2    ⁡  θ  −      sin    2    ⁡  θ    =  1  −  2      sin    2    ⁡  θ    =  2      cos    2    ⁡  θ  −  13)  tan  ⁡  2  θ  =            2      tan      ⁡      θ              1      −              tan        2            ⁡      θ

Bryce Obrien · Accepted Answer

1)  sin  ⁡  2  θ  =  2  sin  ⁡  θ  cos  ⁡  θWe have   sin  ⁡  (  A  +  B  )  =  sin  ⁡  A  ⋅  sin  ⁡  B  +  cos  ⁡  A  ⋅  sin  ⁡  B  A  =  B  =  θ    ⇒  sin  ⁡  2  θ  =  sin  ⁡  θ  ⋅  cos  ⁡  θ  +  cos  ⁡  θ  sin  ⁡  θ    ⇒  sin  ⁡  2  θ  =  2  sin  ⁡  θ  ⋅  cos  ⁡  cos  ⁡  θ  (  p  r  o  v  e  d  )

umm82i · Accepted Answer

2)  cos  ⁡  2  θ  =      cos    2    ⁡  θ  −      sin    2    ⁡  θ    cos  ⁡  (  A  +  B  )  =  cos  ⁡  A  ⋅  cos  ⁡  B  −  sin  ⁡  A  ⋅  sin  ⁡  Blet   A  =  B  =  θ    ⇒  cos  ⁡  2  θ  =      cos    2    ⁡  θ  −      sin    2    ⁡  θ    A  l  s  o      cos    2    ⁡  θ  =  1  −      sin    2    ⁡  θ    ∴  cos  ⁡  2  θ  =  1  −  2      sin    2    ⁡  θ   Proved   .        sin    2    ⁡  θ  =  1  −      cos    2    ⁡  θ        (  F  r  o  m  (  1  )  )    ∴  cos  ⁡  2  θ  =  2      cos    2    ⁡  θ  −  1           (  P  r  o  v  e  d  )3)   tan  ⁡  (  A  +  B  )  =            tan      ⁡      A      +      tan      ⁡      B              1      −      tan      ⁡      A      ⋅      tan      ⁡      B          tan  ⁡  θ  =            2      tan      ⁡      θ              1      −              tan        2            ⁡      θ                 P  r  o  v  e  d

How to prove these identities are true? a)sin(2 theta)=2 sin (theta)cos (theta) b)cos(2 theta)=cos^2 theta-sin^2 theta =1-2 sin^2 theta =2 cos^2 theta-1 c)tan 2theta= (2 tan theta)(1-tan^2 theta)

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