Tan theta / 1- tan theta -cot theta /1-cot theta =cos theta + sin theta/cos theta -sin theta

Answer 1

Brackets are essential here, you probably meant:

TanØ / (1- tan Ø) - cot Ø/(1-cot Ø) = cos Ø + sin Ø/(cos Ø - sin Ø)

check this equation.
What are we doing here, solving? or proving it as an identity?

BTW, I tried it with Ø = 20° , and LS ≠ RS

Answer 2

TanØ / (1- tan Ø) - cot Ø/(1-cot Ø) = (cos Ø + sin Ø)/(cos Ø - sin Ø)

TanØ / (1- tan Ø)
multiply top and bottom by cosØ to get sinØ/(cosØ-sinØ)

similarly, cotØ/(1-cotØ) = cosØ/(sinØ-cosØ)
Now just add them together ...