Question

CosA - sinA + 1 / cosA + sinA - 1 = cosecA + cotA

Answer 1

This is a trig. identity 
(cosA-sinA+1) /(cosA+sinA-1)=cosecA+cotA 

LHS 
= (cosA-sinA+1) /(cosA+sinA-1) 
Multipy the numerator and denominator by (cosA-sinA+1) 

(cosA-sinA+1)^2 /[(cosA+sinA-1)(cosA-sinA+1)] 
= (cos^2A + sin^2A + 1 - 2sinAcosA + 2cosA - 2sinA) / (cos^2A - sin^2A - 2sinA - 1) 
= (2- 2sinAcosA + 2cosA - 2sinA) / (-2sin^2A - 2sinA) 
Divide the numerator and denominator by -2 
(sinAcosA + sinA - cosA - 1) / (sinA(1+sinA)) 
= (sinA(cosA+1) + 1(cosA+1))/(sinA(1+sinA)) 
= (1+sinA)(1+cosA) / (sinA(1+sinA)) 
= (1+cosA)/sinA 
= 1/sinA + cosA/sinA 
= cosecA + cotA 
= RHS

Answer 2

(cosA-sinA+1)^2 /[(cosA+sinA-1)(cosA-sinA+1)]

= (cos^2A + sin^2A + 1 - 2sinAcosA + 2cosA - 2sinA) / (cos^2A - sin^2A - 2sinA - 1)

= (2- 2sinAcosA + 2cosA - 2sinA) / (-2sin^2A - 2sinA)

Divide the numerator and denominator by -2

(sinAcosA + sinA - cosA - 1) / (sinA(1+sinA))

= (sinA(cosA+1) + 1(cosA+1))/(sinA(1+sinA))

= (1+sinA)(1+cosA) / (sinA(1+sinA))

= (1+cosA)/sinA

= 1/sinA + cosA/sinA

= cosecA +