Find Where Undefined/Discontinuous (-1+cot(w)^2+cos(w)^2tan(w)^2)/(csc(w)^2)=cos(w)^4

$\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} = \cos^{4} (w)$

Move $\cos^{4} (w)$ to the left side of the equation by subtracting it from both sides.

$\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Set the denominator in $\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)}$ equal to $0$ to find where the expression is undefined.

$\csc^{2} (w) = 0$

Solve for $w$ .

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Take the $square$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$\csc (w) = \pm \sqrt{0}$

The complete solution is the result of both the positive and negative portions of the solution.

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Simplify the right side of the equation.

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Rewrite $0$ as $0^{2}$ .

$\csc (w) = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$\csc (w) = \pm 0$

$\pm 0$ is equal to $0$ .

$\csc (w) = 0$

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Set the argument in $\cot (w)$ equal to $π n$ to find where the expression is undefined.

$w = π n$ , for any integer $n$

Set the argument in $\tan (w)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$w = \frac{π}{2} + π n$ , for any integer $n$

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

Set-Builder Notation:

${w | w = π n, w = \frac{π}{2} + π n}$

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