Question

tanø+sinø/tanø-sinø= secø+1/secø-1​

Answer 1

sin/cos + sin/1 is certainly NOT 2sin/(cos+1)

any more than 4/3 + 4/1 = 2*4/(3+1) = 2

It just happens that your mistakes eliminated each other.

sin = tan*cos, so

tan+sin = tan(1+cos)

tan-sin = tan(1-cos)

divide and you get (1+cos)/(1-cos)

divide top and bottom by cos and you have

(sec+1)/(sec-1)