Prove the given identities.1. cos 4 ⁡ ( x ) − sin 4 ⁡ ( x ) = cos ⁡ ( 2 x )2. 1 − cos ⁡ ( X ) 1 + cos ⁡ ( X ) = tan 2 ⁡ ( x 2 )

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Prove the given identities.1.      cos          4        ⁡  (  x  )  −      sin          4        ⁡  (  x  )  =  cos  ⁡  (  2  x  )2.            1      −      cos      ⁡      (      X      )              1      +      cos      ⁡      (      X      )        =      tan          2        ⁡  (      x    2    )

Cheyanne Charles · Accepted Answer

1)       cos          4        ⁡  x  −      sin          4        ⁡  x  =  (      cos          2        ⁡  x  −      sin          2        ⁡  x  )  (      cos          2        ⁡  x  +      sin          2        ⁡  x  )we have   (      cos          2        ⁡  x  +      sin          2        ⁡  x  )  =  1     a  n  d         cos          2        ⁡  x  =  2      cos          2        ⁡  x  −  1  =      cos          2        ⁡  x  −      sin          2        ⁡  x  =      cos          2        ⁡  x  −  (  1  −      cos          2        ⁡  x  )  =  2      cos          2        ⁡  x  −  1  =  cos  ⁡  (  2  x  )2) We have the trignometric relations  1  +  cos  ⁡  (  2  t  )  =  2      cos          2        ⁡  t  1  −  cos  ⁡  (  2  t  )  =  2      sin          2        ⁡  tPutting 2t = x, we get  1  +  cos  ⁡  (  x  )  =  2      cos          2        ⁡  (  x      /    2  )  →  1  1  −  cos  ⁡  (  x  )  =  2      sin          2        ⁡  (  x      /    2  )  →  2  2      /    1  ⇒      tan          2        ⁡  (  x      /    2  )

Prove the given identities. 1. cos^4(x)-sin^4(x) = cos(2x) 2. (1-cos(X))/(1+cos(X))=tan^2 (x/2)

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