Question

$sin \beta + cos \beta -1/sin \beta -cos \beta +1= cos \beta /1+sin \beta$

Answer 1

$\frac{sin \beta + cos \beta -1}{sin \beta -cos \beta +1} = \frac{cos \beta} {1+sin \beta }\\ \\ cross-multiply\\ (sin \beta + cos \beta -1)(1+sin \beta) = (sin \beta -cos \beta +1)cos \beta \\ \\ LHS = sin \beta +sin^2 \beta +cos \beta +cos \beta sin \beta -1-sin \beta \\RHS = sin \beta cos \beta -cos^2 \beta +cos \beta \\ \\ LHS = (sin^2 \beta -1) + cos \beta + cos \beta sin \beta = -cos^2 \beta +cos \beta + sin \beta cos \beta \\ \\ = RHS$