Verify the identity: 4 csc(2x) = 2 csc²(x) tan(x)

To verify the identity 4 csc(2x) = 2 csc²(x) tan(x), we will start by manipulating the left-hand side (LHS) and simplifying it to see if it matches the right-hand side (RHS).

Recall that csc(θ) = 1/sin(θ) and that csc(2x) = 1/sin(2x). We can express sin(2x) using the double angle formula: sin(2x) = 2sin(x)cos(x).

Now substitute this into the LHS:

4 csc(2x) = 4 * (1/sin(2x)) = 4 / (2sin(x)cos(x)) = 2 / (sin(x)cos(x))

Next, we know that tan(x) = sin(x)/cos(x), hence csc²(x) = 1/sin²(x) and tan(x) = sin(x)/cos(x). Thus:

RHS = 2 csc²(x) tan(x) = 2 * (1/sin²(x)) * (sin(x)/cos(x)) = 2 * (1/sin(x)cos(x))

Now we can compare LHS and RHS:

LHS = 2 / (sin(x)cos(x)) = RHS

Therefore, because the left-hand side equals the right-hand side, we have verified that:

4 csc(2x) = 2 csc²(x) tan(x)