How can we verify the identity of the equation 4 csc²(2x) = 2 csc²(x) tan(x)?

Verifying the Identity: 4 csc²(2x) = 2 csc²(x) tan(x)

To verify the identity 4 csc²(2x) = 2 csc²(x) tan(x), we need to manipulate one side of the equation to show that it matches the other side. We can start by rewriting both sides using trigonometric identities.

Understanding the Functions

First, let’s recall the definitions of the trigonometric functions involved:

• csc(θ) = 1/sin(θ)
• tan(θ) = sin(θ)/cos(θ)

Manipulating the Left Side

Starting with the left side of the identity:

4 csc²(2x)

We can replace csc²(2x) with 1/sin²(2x), leading to:

4 csc²(2x) = 4 * (1/sin²(2x)) = 4/sin²(2x)

Next, we utilize the identity sin(2x) = 2sin(x)cos(x):

sin²(2x) = (2sin(x)cos(x))² = 4sin²(x)cos²(x)

Substituting this into our equation yields:

4/sin²(2x) = 4/(4sin²(x)cos²(x)) = 1/(sin²(x)cos²(x))