Let's proceed with proving the identity.
We have left-hand side (LHS) as (csc^2X - 1) * sec^2X and right-hand side (RHS) as csc^2X.
Step 1: Start with the expression on the left:
(csc^2X - 1) * sec^2X
Step 2: Now, we know that cscX is the reciprocal of sinX and secX is the reciprocal of cosX. Let's substitute these values in the above expression:
(((1/sinX)^2 - 1) * (1/cosX)^2))
Step 3: Simplify the above expression:
[(1/sin^2X - 1/(sin^2X*cos^2X))]
Step 4: Simplify the above expression to one common denominator:
[(cos^2X - 1)/(sin^2X*cos^2X)]
Step 5: We know that sin^2X + cos^2X = 1. Substitute cos^2X = 1 - sin^2X in above expression:
[(1 - sin^2X - 1)/(sin^2X*cos^2X)]
Step 6: Simplify the above expression to:
[(-sin^2X)/(sin^2X*cos^2X)]
Step 7: Simplify the equation further to reach the RHS of our original equation csc^2X:
1/sin^2X
Therefore, we have proved that
(csc^2X - 1) * sec^2X = csc^2X.