Prove the following Identity: sin 2 ⁡ x − 4 sin ⁡ x + 3 cos 2 ⁡ x = 3 − sin ⁡ x 1 + sin ⁡ x

Question

Prove the following Identity:                     sin                  2                    ⁡      x      −      4      sin      ⁡      x      +      3                      cos                  2                    ⁡      x        =            3      −      sin      ⁡      x              1      +      sin      ⁡      x

Savanah Morton · Accepted Answer

sin                  2                    ⁡      x      −      4      sin      ⁡      x      +      3                      cos                  2                    ⁡      x        =            3      −      sin      ⁡      x              1      +      sin      ⁡      x      left hand side:                    sin                  2                    ⁡      x      −      4      sin      ⁡      x      +      3                      cos                  2                    ⁡      x        =            (      sin      ⁡      x      −      3      )      (      sin      ⁡      x      −      1      )              1      −              sin                  2                    ⁡      x        =            (      sin      ⁡      x      −      3      )      (      sin      ⁡      x      −      1      )              (      1      −      sin      ⁡      x      )      (      1      +      sin      ⁡      x      )        =            −      (      3      −      sin      ⁡      x      )      (      sin      ⁡      x      −      1      )              −      (      sin      ⁡      x      −      1      )      (      1      +      sin      ⁡      x      )        =            3      −      sin      ⁡      x              1      +      sin      ⁡      x        = right hand side

sooxicyiy · Accepted Answer

Cross multiply:  (  sin  ⁡  x  −  3  )  (  sin  ⁡  x  −  1  )  (  1  +  sin  ⁡  x  )  =  (  3  −  sin  ⁡  x  )  (      cos          2        ⁡  x  )This becomes:  (  sin  ⁡  x  −  3  )  (      sin          2        ⁡  x  −  1  )  =  (  3  −  sin  ⁡  x  )  (      cos          2        ⁡  x  ){Since we know that       sin          2        ⁡  x  +      cos          2        ⁡  x  =  1  ,  t  h  e  n     1  −      sin          2        ⁡  x  =      cos          2        ⁡  x  s  o      sin          2        ⁡  x  −  1  =  −      cos          2        ⁡  x}Our equation now becomes:  (  sin  ⁡  x  −  3  )  (  −      cos          2        ⁡  x  )  )  =  (  3  −  sin  ⁡  x  )  (      cos          2        ⁡  x  )This becomes:  (  3  −  sin  ⁡  x  )  (      cos          2        ⁡  x  )  =  (  3  −  sin  ⁡  x  )  (      cos          2        ⁡  x  )

Prove the following Identity: (sin^2 x-4sin x+3)/(cos^2 x)=(3-sin x)/(1+sin x)

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