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

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 the Sum of Two 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 \( 45^{\circ} \) and \( 30^{\circ} \): \[ \tan(75^{\circ}) = \frac{\tan 45^{\circ} + \tan 30^{\circ}}{1 - \tan 45^{\circ} \tan 30^{\circ}} \] ### Step 3: Substitute Known Tangent Values We know: \[ \tan 45^{\circ} = 1 \] \[ \tan 30^{\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \] Substituting these values in: \[ \tan(75^{\circ}) = \frac{1 + \frac{\sqrt{3}}{3}}{1 - 1 \cdot \frac{\sqrt{3}}{3}} \] ### Step 4: Simplify the Expression First, simplify the numerator and denominator: \[ \text{Numerator} = 1 + \frac{\sqrt{3}}{3} = \frac{3 + \sqrt{3}}{3} \] \[ \text{Denominator} = 1 - \frac{\sqrt{3}}{3} = \frac{3 - \sqrt{3}}{3} \] So, \[ \tan(75^{\circ}) = \frac{\frac{3 + \sqrt{3}}{3}}{\frac{3 - \sqrt{3}}{3}} = \frac{3 + \sqrt{3}}{3 - \sqrt{3}} \] ### Step 5: Rationalize the Denominator Multiply the numerator and the denominator by the conjugate of the denominator: \[ \frac{3 + \sqrt{3}}{3 - \sqrt{3}} \times \frac{3 + \sqrt{3}}{3 + \sqrt{3}} = \frac{(3 + \sqrt{3})^2}{3^2 - (\sqrt{3})^2} \] Calculate the squares: \[ (3 + \sqrt{3})^2 = 9 + 6\sqrt{3} + 3 = 12 + 6\sqrt{3} \] \[ 3^2 - (\sqrt{3})^2 = 9 - 3 = 6 \] So, \[ \tan(75^{\circ}) = \frac{12 + 6\sqrt{3}}{6} = 2 + \sqrt{3} \] ### Final Answer \[ \boxed{2 + \sqrt{3}} \]