Find the exact value using a sum or difference identity. \[ \tan 15^{\circ} \] \( [?]-\sqrt{[\square} \) Hint: \( \tan (A \pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)

To find the exact value of \( \tan 15^\circ \) using a sum or difference identity, we can express \( 15^\circ \) as the difference of two angles whose tangent values are known. Let's express \( 15^\circ \) as the difference of \( 45^\circ \) and \( 30^\circ \) since \( \tan 45^\circ = 1 \) and \( \tan 30^\circ = \frac{1}{\sqrt{3}} \). Using the sum or difference identity for tangent: \[ \tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \] Substitute \( A = 45^\circ \) and \( B = 30^\circ \) into the formula: \[ \tan 15^\circ = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ} \] Now, we can calculate the exact value of \( \tan 15^\circ \) using the known values of \( \tan 45^\circ \) and \( \tan 30^\circ \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(1-\frac{1}{\sqrt{3}}\right)}{\left(1+1\times \left(\frac{1}{\sqrt{3}}\right)\right)}\) - step1: Remove the parentheses: \(\frac{1-\frac{1}{\sqrt{3}}}{1+1\times \left(\frac{1}{\sqrt{3}}\right)}\) - step2: Remove the parentheses: \(\frac{1-\frac{1}{\sqrt{3}}}{1+1\times \frac{1}{\sqrt{3}}}\) - step3: Multiply: \(\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}\) - step4: Subtract the numbers: \(\frac{\frac{3-\sqrt{3}}{3}}{1+\frac{1}{\sqrt{3}}}\) - step5: Add the numbers: \(\frac{\frac{3-\sqrt{3}}{3}}{\frac{3+\sqrt{3}}{3}}\) - step6: Multiply by the reciprocal: \(\frac{3-\sqrt{3}}{3}\times \frac{3}{3+\sqrt{3}}\) - step7: Reduce the numbers: \(\left(3-\sqrt{3}\right)\times \frac{1}{3+\sqrt{3}}\) - step8: Multiply: \(\frac{3-\sqrt{3}}{3+\sqrt{3}}\) - step9: Multiply by the Conjugate: \(\frac{\left(3-\sqrt{3}\right)\left(3-\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\) - step10: Multiply the numbers: \(\frac{\left(3-\sqrt{3}\right)\left(3-\sqrt{3}\right)}{6}\) - step11: Multiply the numbers: \(\frac{12-6\sqrt{3}}{6}\) - step12: Rewrite the expression: \(\frac{6\left(2-\sqrt{3}\right)}{6}\) - step13: Reduce the fraction: \(2-\sqrt{3}\) The exact value of \( \tan 15^\circ \) is \( 2 - \sqrt{3} \).