\operatorname { cosec } ^ { 6 } A - \operatorname { cot } ^ { 6 } A =

1.	\operatorname { cosec } ^ { 6 } A - \operatorname { cot } ^ { 6 } A = 3 \operatorname { cot } ^ { 2 } A \operatorname { cosec } ^ { 2 } A + 1
Answer» cosec^6 A = cot^6 A + 3cot^2 A cosec^2 A + 1 Then, we will prove it as : To Prove: cosec^6 A = cot^6 A + 3 cot^2 A cosec^2 A + 1 i.e. cosec^6A – cot^6A – 3 cot^2 A cosec^2A = 1 Taking L.H.S. , cosec^6A – cot^6A – 3 cot^2 A cosec^2 A =(cosec^2 A)^3– (cot^2 A)^3– 3 cot^2 A cosec^2 A ={(cosec^2 A – cot^2 A) ((cosec^2 A)^2+ (cot^2 A)^2+ cosec^2 A cot^2 A)} ={1 ((cosec^2 A)^2+ (cot^2 A)^2– 2 cosec^2 A cot^2 A + 2 cosec^2 A cot^2 A + cosec^2A cot^2 A)} – 3 cot^2 A cosec^2 A ={(cosec^2 A – cot^2 A)^2+ 3 cosec^2Acot^2A} – 3 cosec^2A cot^2 A =(cosec^2 A – cot^2 A)^2 =(1)2 = 1 Hence, cosec^2 A – cot^6A – 3 cot^2A cosec^2 A = 1 ⇒ cosec^2 A -cot^6 A =3 cot^2A cosec^2A + 1 thats not the question

\operatorname { cosec } ^ { 6 } A - \operatorname { cot } ^ { 6 } A = 3 \operatorname { cot } ^ { 2 } A \operatorname { cosec } ^ { 2 } A + 1

Answer»

cosec^6 A = cot^6 A + 3cot^2 A cosec^2 A + 1

Then, we will prove it as :

To Prove:

cosec^6 A = cot^6 A + 3 cot^2 A cosec^2 A + 1

i.e. cosec^6A – cot^6A – 3 cot^2 A cosec^2A = 1

Taking L.H.S. ,

cosec^6A – cot^6A – 3 cot^2 A cosec^2 A

=(cosec^2 A)^3– (cot^2 A)^3– 3 cot^2 A cosec^2 A

={(cosec^2 A – cot^2 A) ((cosec^2 A)^2+ (cot^2 A)^2+ cosec^2 A cot^2 A)}

={1 ((cosec^2 A)^2+ (cot^2 A)^2– 2 cosec^2 A cot^2 A + 2 cosec^2 A cot^2 A + cosec^2A cot^2 A)} – 3 cot^2 A cosec^2 A

={(cosec^2 A – cot^2 A)^2+ 3 cosec^2Acot^2A} – 3 cosec^2A cot^2 A

=(cosec^2 A – cot^2 A)^2

=(1)2

= 1

Hence,

cosec^2 A – cot^6A – 3 cot^2A cosec^2 A = 1

⇒ cosec^2 A -cot^6 A =3 cot^2A cosec^2A + 1

thats not the question