ICot (A+45) - tan(A-45) = 2cos2A/1+sin2A

Question

Prove it please

Answer 1

 

cot(A+45)=tan(90-(A+45))=tan(45-A); tan(A-45)=-tan(45-A).

The given expression can be written tan(45-A)+tan(45-A)=2tan(45-A).

2tan(45-A)=2[(tan(45)-tan(A))/(1+tan(45)tan(A))]=2[(1-tan(A))/(1+tan(A))].

Multiply top and bottom by cos(A): 2[(cos(A)-sin(A))/(cos(A)+sin(A))].

Multiply top and bottom by the denominator:

2[(cos²(A)-sin²(A))/(cos²(A)+2sin(A)cos(A)+sin²(A))]=2cos(2A)/(1+sin(2A)) QED