4.2 Hence calculate the value of \( \tan 75^{\circ} \) without using a calculator.

To calculate \( \tan 75^{\circ} \) without using a calculator, we can use the **angle addition formula** for tangent. Here's a step-by-step method: ### Step 1: Express 75° as the Sum of Two Known Angles Choose angles whose tangent values are known. A convenient choice is: \[ 75^{\circ} = 45^{\circ} + 30^{\circ} \] ### Step 2: Use the Tangent Addition Formula The tangent addition formula is: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] Applying this to \( A = 45^{\circ} \) and \( B = 30^{\circ} \): \[ \tan(75^{\circ}) = \frac{\tan 45^{\circ} + \tan 30^{\circ}}{1 - \tan 45^{\circ} \tan 30^{\circ}} \] ### Step 3: Substitute Known Tangent Values We know that: \[ \tan 45^{\circ} = 1 \quad \text{and} \quad \tan 30^{\circ} = \frac{1}{\sqrt{3}} \] Substituting these values: \[ \tan(75^{\circ}) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \] ### Step 4: Simplify the Expression Multiply numerator and denominator by \( \sqrt{3} \) to eliminate the radicals: \[ \tan(75^{\circ}) = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \] ### Step 5: Rationalize the Denominator Multiply the numerator and denominator by the conjugate \( (\sqrt{3} + 1) \): \[ \tan(75^{\circ}) = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} \] ### Step 6: Final Simplification Divide numerator and denominator by 2: \[ \tan(75^{\circ}) = 2 + \sqrt{3} \] ### **Conclusion** \[ \tan 75^{\circ} = 2 + \sqrt{3} \]