Question

Prove that Cosec6A-Cot6A=1+3Cosec2A.Cot2A

Answer 1

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 cosec2A

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

= (1)2

=  1

Hence,

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

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

hope answer helpful