Solve the equation #2cos^2xcotx=cotx# within the interval #0<=x<2pi#?
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To solve the equation (2\cos^2(x) \cot(x) = \cot(x)) within the interval (0 \leq x < 2\pi), follow these steps:

Rewrite (2\cos^2(x) \cot(x)) as (2\cos(x)).

Set up the equation: (2\cos(x) = \cot(x)).

Rewrite (\cot(x)) as (\frac{\cos(x)}{\sin(x)}).

Now the equation becomes (2\cos(x) = \frac{\cos(x)}{\sin(x)}).

Multiply both sides by (\sin(x)) to clear the fraction.

We get (2\cos(x) \sin(x) = \cos(x)).

Rewrite (\sin(x)) as (1  \cos^2(x)) using the Pythagorean identity.

So, the equation becomes (2\cos(x)(1  \cos^2(x)) = \cos(x)).

Distribute and rearrange terms: (2\cos(x)  2\cos^3(x) = \cos(x)).

Move all terms to one side of the equation: (2\cos(x)  2\cos^3(x)  \cos(x) = 0).

Combine like terms: (2\cos(x)  \cos(x)  2\cos^3(x) = 0).

Simplify further: (\cos(x)(2  1  2\cos^2(x)) = 0).

Set each factor equal to zero:
 (\cos(x) = 0)
 (2  1  2\cos^2(x) = 0)
 Solve each equation separately:
 For (\cos(x) = 0), (x = \frac{\pi}{2}, \frac{3\pi}{2}).
 For (2  1  2\cos^2(x) = 0), solve for (\cos(x)), which gives (\cos(x) = \pm \frac{\sqrt{2}}{2}). The corresponding angles are (x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}).
 Check the solutions in the original equation to ensure they are valid within the given interval (0 \leq x < 2\pi).
After checking, the valid solutions within the interval (0 \leq x < 2\pi) are:
[x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{4}, \frac{7\pi}{4}]
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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