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Chapter 5: Problem 101

Explain what happens when you divide each side of the equation \(\cot x \cos ^{2} x=2 \cot x\) by cot \(x .\) Is this a correct method to use when solving equations?

Short Answer

Expert verified
By dividing each side by \(\cot x\), the given equation simplifies to \(\cos ^{2} x = 2\). However, it is important to note that this method may potentially exclude solutions, as we can only divide by \(\cot x\) if \(\cot x\neq 0\).

Step by step solution

01

Understand the Equation

The given equation is \(\cot x \cos ^{2} x = 2 \cot x\) The objective is to divide each side of this equation by \(\cot x\).
02

Dividing Left Side by \(\cot x\)

When \(\cot x \cos ^{2} x\) is divided by \(\cot x\), the \(\cot x\) in the denominator cancels out with the \(\cot x\) in the numerator leaving only \(\cos ^{2} x\) as result.
03

Dividing Right Side by \(\cot x\)

On the right side of the equation, when \(2 \cot x\) is divided by \(\cot x\), the \(\cot x\) terms cancel out exactly as on the left side, leaving 2 as result.
04

Checking if this is a Correct Method

Dividing by \(\cot x\) is valid as long as \(\cot x\) is not equal to zero. Also it's important to remember that even though the simplification makes the equation appear easier to solve, the original equation might have solutions which would be affected when we divide by a variable terms (in this case \(\cot x\)). Thus, always ensure that the variable you are dividing by is not zero.

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