Draw a triangle in the plane with vertices <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>,</mo><msqrt><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt><mo>)</mo></mrow></mstyle></math> , <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> , and the origin. Then <math><mstyle displaystyle="true"><mi>arccos</mi><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> is the angle between the positive x-axis and the ray beginning at the origin and passing through <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>,</mo><msqrt><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt><mo>)</mo></mrow></mstyle></math> . Therefore, <math><mstyle displaystyle="true"><mi>tan</mi><mrow><mo>(</mo><mi>arccos</mi><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>6</mn><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mrow><mrow><mn>6</mn><mi>x</mi></mrow></mfrac></mstyle></math> .

Since both terms are perfect squares, factor using the difference of squares formula, <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>i</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>a</mi><mo>-</mo><mi>i</mi><mo>)</mo></mrow></mstyle></math> where <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>i</mi><mo>=</mo><mn>6</mn><mi>x</mi></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

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