First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, <math><mstyle displaystyle="true"><mn>15</mn><mi>°</mi></mstyle></math> can be split into <math><mstyle displaystyle="true"><mn>60</mn><mo>-</mo><mn>45</mn></mstyle></math> .

Use the difference formula for sine to simplify the expression. The formula states that <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>A</mi><mo>-</mo><mi>B</mi><mo>)</mo></mrow><mo>=</mo><mi>sin</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>60</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>45</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>60</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>45</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mo>(</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Combine using the product rule for radicals.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Combine the numerators over the common denominator.

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

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