Question

Use trigonometric identity.....
cosec^ 6 A - cot^6A = 3cot^2Acosec^2A +1
Prove this

Answer 1

Answer:

Step-by-step explanation:

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

Then, we will prove it as :

To Prove:

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

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

Taking L.H.S. ,

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

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

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

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

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

= (cosec ^2A – cot^ 2A)^2 + 0

= (1)^2

=  1

Hence,

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

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

=> cosec^ 6 A - cot^6A = 3cot^2Acosec^2A +1