prove cosec2A+ cot4A = cotA - cosec4A

Answer :

We have : Cosec ² A + Cot ⁴ A = Cosec ⁴ A - Cot ² A

Taking LHS  :

$\Rightarrow$Cosec ² A + Cot ⁴ A

$\Rightarrow$Cosec ² A + ( Cot ² A )² 

$\Rightarrow$Cosec ² A + ( Cosec ² A - 1 )²                                         (  we know : 1 + Cot ² $\mathrm{\theta }$ =  Cosec ² $\mathrm{\theta }$ )

$\Rightarrow$Cosec ² A +  Cosec ⁴ A  + 1  - 2 Cosec ² A

$\Rightarrow$ Cosec ⁴ A  + 1  - Cosec ² A

$\Rightarrow$ Cosec ⁴ A  + 1  - ( 1 + Cot ² A)

$\Rightarrow$ Cosec ⁴ A  + 1  -  1 - Cot ² A

$\Rightarrow$ Cosec ⁴ A  - Cot ² A

LHS  =  RHS                                          ( Hence proved )