Question

Prove that $\frac{cos\theta-sin\theta+1}{cos\theta+sin\theta-1} = cosec\theta+cot\theta$

Answer 1

i give you an outline of the solution.

multiply LHS  by    cos + (1- sin )    in numerator and denom.

LHS = (cos² + 1 + sin² - 2 sin + 2 cos - 2 cos sin) / (cos² - 1 - sin² + 2 sin)
       = (1 - sin + cos - sin cos)/ (sin - sin²)
       = (1 - sin) (1+cos)/(1- sin)sin
       = cosec + cot