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Chapter 5: Problem 34

Perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer. $$(2 \csc x+2)(2 \csc x-2)$$

Short Answer

Expert verified
The simplified form of the given expression is \(4 \cot^2 x\).

Step by step solution

01

Identify the Form

The given expression \((2 \csc x+2)(2 \csc x-2)\) is a perfect square difference, which is of the form \(a^2 - b^2\), where \(a = 2 \csc x\) and \(b = 2\).
02

Apply the identity for square difference

The formula for a square difference can be applied here: \(a^2 - b^2 = (a+b)(a-b)\). Therefore, \((2 \csc x+2)(2 \csc x-2) = (2 \csc x)^2 - 2^2\).
03

Simplify using trigonometric identity

The result obtained from above is \(4 \csc^2 x - 4\). Now, the Pythagorean identity for cosecant, \(\csc^2 x = 1 + \cot^2 x\), can be used to simplify this further, giving \(4(1 + \cot^2 x) - 4\).
04

Simplify

Finally, simplify the above expression to get the final simplified form: \(4 \cot^2 x\).

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