Problem 27 - Precalculus Essentials

Question

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\frac{\tan 10^{\circ}+\tan 35^{\circ}}{1-\tan 10^{\circ} \tan 35^{\circ}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\tan 45^{\circ}\), which has an exact value of 1

Step 1: Identify Formula Structure

Check whether the given expression fits into the form of the sum formula for tangents. The given expression takes the form \(\frac{\tan A + \tan B}{1 - \tan A \tan B}\), which matches the formula for \(\tan (A + B)\). So, this can be written as \(\tan (10^{\circ} + 35^{\circ})\).

Step 2: Simplify The Expression

Solve the addition inside the tangent function: \(10^{\circ} + 35^{\circ} = 45^{\circ}\). Now, our expression looks like this: \(\tan 45^{\circ}\).

Step 3: Find The Exact Value

The tangent of 45 degrees is 1. So, the given expression has an exact value of 1.