Can someone help me with this trigonometry Identity problem?

These are fundamental trigonometric identities to use:

sin^2 x +cos^2 x = 1

sin(2x) = 2sinx cosx

cos(2x) = cos^2 x -sin^2x

sin(x+y) = sinx cosy +siny cosx

cos(x+y) = cosx cosy - sinx siny

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[sin(3x)+cos(3x)]/(cosx -sinx) = 1+2sin(2x)

Work on the L.H.S. and multiply by the conjugate of the deonominator

[sin(3x)+cos(3x)](cosx +sinx) /(cos^2 x -sin^2 x) = R.H.S.

Simplify the denominator

[sin(3x)(cosx +sinx) +cos(3x)(cosx +sinx)] /cos(2x) = R.H.S.

Expand sin(3x) and cos(3x)

[(sin(2x)cosx +sinx cos(2x))(cosx +sinx) +(cos(2x)cosx -sin(2x)sinx)(cosx +sinx)] /cos(2x) = R.H.S.

Multiply everything out and cancel the terms +sin(2x)sinx cosx and -sin(2x)sinx cosx

[sin(2x)cos^2 x +sinx cosx cos(2x) +sin^2 x cos(2x) +cos(2x)cos^2 x +cos(2x)sinx cosx -sin^2 x sin(2x)] /cos(2x) = R.H.S.

Can you finish the problem from here?

[cos(2x)(cos^2 x +sin^2 x +2sinx cosx) +sin(2x)(cos^2 x -sin^2 x)] /cos(2x) = R.H.S.

[cos(2x)(1 +sin(2x)) +sin(2x)cos(2x)] /cos(2x) = R.H.S.

The identity has been verified by showing the left hand side can be written as the right hand side

1 +2sin(2x) = 1 +2sin(2x)

     Convert to ONLY single angle trig functions by substituting identities.

Identities used:

sin(3x) = - 4sin³x + 3sinx

cos(3x) =  4cos³x – 3cosx

sin(2x) = 2sinx  •  cosx

sin²x + cos²x = 1

  (- 4sin³x + 3sinx + 4cos³x – 3cosx) / (cosx – sinx)  =  1 + 4sinx cosx

        ... multiply by: (cosx – sinx)

  - 4sin³x + 3sinx + 4cos³x – 3cosx  =  (cosx – sinx) (1 + 4sinx cosx)

        ... expand

  - 4sin³x + 3sinx + 4cos³x – 3cosx  =  cosx  +  4sinx cos²x – sinx  –  4sin²x cosx

        ... combine like terms

  - 4sin³x + 4sinx + 4cos³x – 4cosx  =  4sinx cos²x  –  4sin²x cosx

  - sin³x + sinx + cos³x – cosx  =  sinx cos²x  –  sin²x cosx

        ... factor left side twice

  sinx(- sin²x + 1)  +  cosx(cos²x – 1)  =  sinx cos²x  –  sin²x cosx

        ... rearranging terms

  sinx(1 – sin²x)  –  cosx(1 – cos²x)  =  sinx cos²x  –  sin²x cosx

  sinx cos²x – sin²x cosx  =  sinx cos²x  –  sin²x cosx   TRUE