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Chapter 5: Problem 73

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 \(4\cot^2x\).

Step by step solution

01

Use the binomial formula

Recognize that the given expression follows the pattern of a difference of squares formula \(a^2-b^2=(a+b)(a-b)\). Substituting \(a\) for \(2 \csc x\) and \(b\) for \(2\), the multiplication becomes \((2 \csc x)^2-(2)^2\). Apply this formula.
02

Compute the squares and difference

Compute the squares and their difference, \(a^2-b^2\), to simplify the given expression. This leads to \(4 \csc^2 x - 4\).
03

Use the Pythagorean identity

Use the Pythagorean identity for cosecant, which states that \(\csc^2 x = 1+ \cot^2 x\), to simplify the expression further. Substitute this identity into the equation: \(4 \csc^2 x - 4\) becomes \(4(1+ \cot^2 x) - 4\).
04

Simplify the expression

Apply the operations to the elements inside and outside the parenthesis. This simplifies the expression to \(4 + 4 \cot^2 x - 4\). The positive and negative 4 will cancel out, leaving you with a final simplified expression, which is \(4\cot^2x\).

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