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

To calculate \( \tan 75^{\circ} \) without using a calculator, we can use the **tangent addition formula**. Here's a step-by-step breakdown: ### Step 1: Express \( 75^{\circ} \) as a Sum of Known Angles We can write: \[ 75^{\circ} = 45^{\circ} + 30^{\circ} \] ### Step 2: Apply 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 Values We know that: \[ \tan(45^{\circ}) = 1 \quad \text{and} \quad \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \] Substituting these values in: \[ \tan(75^{\circ}) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \] ### Step 4: Rationalize the Denominator To simplify \( \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \), multiply the numerator and the denominator by the conjugate \( \sqrt{3} + 1 \): \[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \] ### Final Answer \[ \tan 75^{\circ} = 2 + \sqrt{3} \]