Final answer:
To prove the given trigonometric identity, we simplify both sides of the equation and show that they are equal. We start by simplifying the left side and then substitute an identity to simplify it further. Finally, we show that the left side is equal to 2 sec^2 A, which implies that the identity is proven.
Step-by-step explanation:
To prove cosec A/cosec A - 1 + cosec A/cosec A + 1 = 2 sec^2 A, we need to simplify both sides of the equation and show that they are equal.
Starting with the left side, we can find a common denominator by multiplying the first fraction by (cosec A + 1)/(cosec A + 1), and the second fraction by (cosec A - 1)/(cosec A - 1). This gives us (cosec A * (cosec A + 1) + cosec A * (cosec A - 1)) / ((cosec A - 1) * (cosec A + 1)).
Simplifying further, we have (cosec^2 A + cosec A + cosec^2 A - cosec A) / (cosec^2 A - 1), which simplifies to (2 cosec^2 A) / (cosec^2 A - 1).
Next, we can use the identity sec^2 A = 1 + cosec^2 A, which means cosec^2 A = sec^2 A - 1. Substituting this into our expression, we get (2 cosec^2 A) / (sec^2 A - 1), which is equal to 2 sec^2 A.