Question

if cosec A-cot A =√5then prove that cosec A + cot A =3

Answer 1

Given,

cosec A-cot A =√5
Making square of both sides
(cosec A-cot A)² =5
cosec²A + cot²A -2 cosecA.cotA. = 5
1+ cot²A + cot²A -2 cosecA.cotA.= 5
2cot²A -2 cosecA.cotA = 4
cot²A - cosecA.cotA= 2
cotA(cotA - cosec A) = 2
cotA(-√5) = 2
cotA = -2/√5
cot²A = 4/5
Now,
We know that ,

cosec²A = 1+cot²A

cosec²A= 1+ 4/5

cosec²A = (5+4)/5

cosec²A= 9/5
cosecA = 3/√5
Now,
cosecA + cotA = 3/√5 - 2/√5
cosecA+ cotA = 1/√5.
Hence question should be correct as:-

if cosec A-cot A =√5then prove that cosec A + cot A =1/√5