First check out the proposed identity by substituting for A. Let A=30°, sinA=½, sin⁴A=1/16, cosA=√3/2, cos⁴A=9/16, tan(2A)=√3, cos(2A)=½.
Then, 1/16+√3/2-9/16=(√3-1)/2≠½. The identity is false.
Now, let’s see if we can reduce the expressions.
sin⁴A-cos⁴A=(sin²A-cos²A)(sin²A+cos²A)=-(cos²A-sin²A)(1).
cos²A-sin²A=cos(2A).
2sinAcosA=sin(2A).
So we have to prove:
(sin(2A)-cos(2A))/tan(2A)=cos(2A).
Multiply through by tan(2A):
sin(2A)-cos(2A)=sin(2A).
This is clearly false as an identity, because it implies -cos(2A)=0.
Even if the proposed identity was:
sin⁴A+2sinAcosA-(cos⁴A/tan(2A))=cos(2A), it’s still false.