Question

Prove that cos theta - sin theta + 1/ cos theta + sin theat - 1 = cosec theta + cot theta

Answer 1

cosθ - sinθ - 1 / cosθ + sinθ - 1  
 dividing both numerator and denominator by sinθ

cotθ - 1 + cosecθ / cotθ + 1 -cosecθ   
= cotθ - (cosec²θ - cot²θ) + cosecθ / cotθ + 1 - cosecθ 
= cotθ + cosecθ - (cosecθ - cotθ)(cosecθ + cotθ) / cotθ - cosecθ + 1)
= (cosecθ + cotθ)( 1 - cosecθ + cotθ) / (1 - cosecθ + cotθ) = cosecθ + cotθ

hence proved