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

To calculate \( \tan 75^{\circ} \) without a calculator, we can use the angle sum identity for tangent: \[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \] We can express \( 75^{\circ} \) as \( 45^{\circ} + 30^{\circ} \). So, we have: \[ \tan 75^{\circ} = \tan(45^{\circ} + 30^{\circ}) = \frac{\tan 45^{\circ} + \tan 30^{\circ}}{1 - \tan 45^{\circ} \tan 30^{\circ}} \] Knowing that \( \tan 45^{\circ} = 1 \) and \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \), we substitute these values in: \[ \tan 75^{\circ} = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \] Simplifying the numerator: \[ = \frac{\frac{\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} \] Canceling \(\frac{1}{\sqrt{3}}\): \[ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \] To make it cleaner, multiply the numerator and denominator by \( \sqrt{3} + 1 \) (the conjugate): \[ = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \] Thus, the value of \( \tan 75^{\circ} \) is: \[ \boxed{2 + \sqrt{3}} \]