Question

Cot^2A/cosecA-1 -1 = cosecA

Answer 1

I assume the Question is to prove

$\rm\dfrac{\cot^2 A}{cosec \ A -1} -1=cosec \ A$

Solution:-

We know that,

cosec² A – cot² A = 1

Therefore,

cot² A = cosec ² A - 1

Substituting the value of cot² A

$\rm\dashrightarrow\dfrac{cosec^2 \ A-1}{cosec \ A -1} -1=cosec \ A$

Since, a² - b² = (a-b)(a+b)

∴

$\rm\dashrightarrow\dfrac{cosec^2 \ A-1^2}{cosec \ A -1} -1=cosec \ A$

$\rm\dashrightarrow\dfrac{(cosec \ A -1)(cosec \ A +1)}{cosec \ A -1} -1=cosec \ A$

(cosec A - 1)gets cancelled on both the numerator and denominator

$\rm\dashrightarrow cosec \ A +1 -1=cosec \ A$

$\rm\dashrightarrow cosec \ A =cosec \ A$

Hence, Proved.