Question

cosec^4A - cosec^2A = cot^4 A+ cot^2A​

Answer 1

Step-by-step explanation:

Solution :-

LHS :-

cosec^4A - cosec^2A

=> Cosec^2 A × Cosec^2 A - Cosec^2 A

=> Cosec^2 A (Cosec^2 A - 1)

We know that

Cosec^2 A - Cot^2 A = 1

Cot^2 A = Cosec^2 A -1 and

Cosec^2 A = 1+Cot^2 A

=> Cosec^2 A ( Cot^2 A)

=> (1+Cot^2 A) (Cot^2 A)

=> Cot^2 A + Cot^2 A× Cot^2 A

=> Cot^2 A + Cot^4 A

=> RHS

LHS = RHS

or

RHS :

Cot^2 A + Cot^4 A

=> Cot^2A(1+Cot^2A)

We know that

Cosec^2 A - Cot^2 A = 1

Cot^2 A = Cosec^2 A -1 and

1+Cot^2 A = Cosec^2 A

=> (Cosec^2 A -1)(Cosec^2 A)

=> cosec^2 A× Cosec^2 A - Cosec^2 A

=> Cosec^4 A - Cosec^2 A

=> LHS

RHS = LHS

Hence, Proved.

Used formulae:-

• Cosec^2 A - Cot^2 A = 1
• Cot^2 A = Cosec^2 A -1

• Cosec^2 A = 1+Cot^2 A