Question

find the exact value using a sum or difference identity. tan 75° hint: tan(a ± b) = \frac{tan a ± tan b}{1 ∓ tan a tan b}

Explanation:

Step1: Express \(75^{\circ}\) as a sum

Express \(75^{\circ}\) as \(45^{\circ}+30^{\circ}\). So, \(\tan75^{\circ}=\tan(45^{\circ} + 30^{\circ})\).

Step2: Apply the tangent sum identity

The tangent sum identity is \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\). Here \(A = 45^{\circ}\) and \(B=30^{\circ}\).
We know that \(\tan45^{\circ}=1\) and \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\).
Substitute these values into the formula:

$$
LATEXBLOCK0
$$

Step3: Rationalize the denominator

Multiply the numerator and denominator by \(3 + \sqrt{3}\):

$$
LATEXBLOCK1
$$

Answer:

\(2+\sqrt{3}\)