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Chapter 5: Problem 31

Factor the trigonometric expression. There is more than one correct form of each answer. $$\cot ^{2} x+\csc x-1$$

Short Answer

Expert verified
The factored form of the given expression is \((\csc x -1)^2\)

Step by step solution

01

Step 1

Utilize the trigonometric identity \(\cot ^{2} x = \csc ^{2} x - 1\) to replace \(\cot ^{2} x\) in the given expression. The expression can then be rewritten to \((\csc ^{2} x - 1)+\csc x-1\)
02

Step 2

Next simplify the expression by combining like terms. So it becomes \(2\csc ^{2} x-1 -\csc x\)
03

Step 3

Now you can factor by grouping: Group the first two terms together and the last term separately. This will give \( (\csc x - 1)(2\csc x - 1)\)
04

Step 4

The above factored form is a correct answer. However, more simplified forms could also be found by factoring out a \(\csc x\) from the first group to form \(\csc x(\csc x - 1) - 1(\csc x-1)\) which can further be simplified to \((\csc x -1)^2\)

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