One more trigonometry question?

Question

cot A+ cosec A-1

------------------------ = cosec A + cot A

cot A-cosec A +1

Answer 1

LHS, Numerator:

cot A+ cosec A -1 =  cotA + cosecA - (cosec²A - cot²A)

[since sin²A + cos²A = 1 => 1- cos²A = sin²A

=>Divide by sin²A both sides, cosec²A - cot²A = 1]

  = cotA + cosecA - (cosecA + cotA)(cosecA - cotA)

  = (cosecA + cotA) (cotA - cosecA + 1) [taking the common factor cosecA + cotA]

Hence LHS:

(cosecA + cotA) (cotA - cosecA + 1) / (cotA - cosecA + 1)

= cosecA + cotA = RHS

AJM

Answer 2

now if i start like this....

cotA+cosecA-1 = cotA +cosecA-(cosecA^2-CotA^2)

                        = (cotA + cosecA)-(cosecA-cotA)(cosecA+cotA)

                        = (cosecA+cotA) [1-(cosecA-cotA)]

                        = (cosecA+cotA)(1-cosecA + cotA)

so it's

cot A+ cosec A-1

------------------------ = cosec A + cot A

cot A-cosec A +1

Answer 3

cot A+ cosec A-1

= cot A + cosec A - (cosec^2 A - cot^2 A)

= cot A + cosec A - (cosec A + cot A)(cosec A - cot A)

= (cot A + cosec A) ( 1- cosec A + 1)

hence

(cot A+ cosec A-1)/( 1- cosec A + 1) = cot A + cosec A