2. (5 pts) demonstrate use of an appropriate sum or difference formula to determine the exact value of…

Explanation:

Step 1: Decompose the angle using sum formula

Express (285^\circ) as (240^\circ + 45^\circ), where (240^\circ = 180^\circ + 60^\circ) and (\tan(240^\circ) = \tan(60^\circ) = \sqrt{3}) (since tangent is positive in the third quadrant).

Step 2: Apply the tangent sum formula

Use (\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}):
[ \tan(285^\circ) = \tan(240^\circ + 45^\circ) = \frac{\tan 240^\circ + \tan 45^\circ}{1 - \tan 240^\circ \tan 45^\circ} = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} ]

Step 3: Rationalize the denominator

Multiply numerator and denominator by (1 + \sqrt{3}):
[ \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{3 + 2\sqrt{3} + 1}{1 - 3} = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3} ]

Answer:

(-2 - \sqrt{3})