Question

Prove the following identities:

cosec⁴ A (1 – cos⁴ A) – 2 cot² A = 1

Answer 1

cosec⁴ A (1 – cos⁴ A) – 2 cot² A

= cosec⁴ A (1 – cos² A) (1 + cos² A) – 2 cot² A

= cosec⁴ A (sin² A) (1 + cos² A) – 2 cot² A

= cosec² A (1 + cos² A) – 2 cot² A

= ${\displaystyle \cos e{c}^{2}A+\frac{{\cos }^{2}A}{{\sin }^{2}A}-2{\cot }^{2}A}$

= cosec² A + cot² A – 2 cot² A

= cosec² A – cot² A

= 1