Saw me secA=x2+1/x2-1

Question

CosecA+cotA=x CosecA-cotA=1/x (given)

Answer 1

x^2+1=(1+cosA)^2/sin^2A+1=

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

2(1+cosA)/sin^2A; (1)

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

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

2cosA(1+cosA)/sin^2A. (2)

(x^2+1)/(x^2-1)=(1)/(2)=2(1+cosA)/(2cosA(1+cosA))=secA QED

Also note that cosecA+cotA=(1+cosA)/sinA and 1/x=(1-cosA)/sinA

Multiply these together: (1-cos^2A)/sin^2A=sin^2A/sin^2A=x/x=1, so the two given equations are equivalent.

Or (1+cosA)/sinA=sinA/(1-cosA), because 1-cos^2A=sin^2A when we cross-multiply, substituting x=cosecA+cotA and 1/x=cosecA-cotA, i.e., x=1/(cosecA-cotA)=sinA/(1-cosA).