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

Move <math><mstyle displaystyle="true"><msup><mi>cos</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mstyle></math> to the left side of the equation by subtracting it from both sides.

Set the denominator in <math><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mn>1</mn><mo>+</mo><msup><mi>cot</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><msup><mi>tan</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow><mrow><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the expression is undefined.

Take the <math><mstyle displaystyle="true"><mtext class="not-bold-word">square</mtext></mstyle></math> root of both sides of the <math><mstyle displaystyle="true"><mtext class="not-bold-word">equation</mtext></mstyle></math> to eliminate the exponent on the left side.

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

Simplify the right side of the equation.

Rewrite <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

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

The range of cosecant is <math><mstyle displaystyle="true"><mi>y</mi><mo>≤</mo><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>y</mi><mo>≥</mo><mn>1</mn></mstyle></math> . Since <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> does not fall in this range, there is no solution.

No solution

No solution

Set the argument in <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mstyle></math> equal to <math><mstyle displaystyle="true"><mi>π</mi><mi>n</mi></mstyle></math> to find where the expression is undefined.

Set the argument in <math><mstyle displaystyle="true"><mi>tan</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mstyle></math> equal to <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> to find where the expression is undefined.

The equation is undefined where the denominator equals <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the argument of a square root is less than <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , or the argument of a logarithm is less than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Set-Builder Notation:

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