Question

Prove that \(\tan 7^\circ \cdot \tan 60^\circ \cdot \tan 83^\circ = \sqrt{3}\).

Hint:

In trigonometric products or sums, always check for pairs of angles that add up to 90\(^\circ\) or 180\(^\circ\). This is a strong hint to use complementary or supplementary angle identities.

Answer 1

Step 1: Understanding the Concept:
This problem requires the use of trigonometric identities, specifically the relationship between trigonometric functions of complementary angles (angles that sum to 90\(^\circ\)).

Step 2: Key Formula or Approach:
The key identities are: 1. \(\tan(90^\circ - \theta) = \cot\theta\) 2. \(\tan\theta \cdot \cot\theta = 1\) We also need the standard value of \(\tan 60^\circ = \sqrt{3}\).

Step 3: Detailed Explanation:
Let's start with the Left Hand Side (LHS) of the equation: \[ \text{LHS} = \tan 7^\circ \cdot \tan 60^\circ \cdot \tan 83^\circ \] Notice that \(7^\circ + 83^\circ = 90^\circ\). This means they are complementary angles. We can rewrite \(\tan 83^\circ\) using the complementary angle identity: \[ \tan 83^\circ = \tan(90^\circ - 7^\circ) = \cot 7^\circ \] Now substitute this back into the LHS expression: \[ \text{LHS} = \tan 7^\circ \cdot \tan 60^\circ \cdot \cot 7^\circ \] Rearrange the terms to group the complementary functions together: \[ \text{LHS} = (\tan 7^\circ \cdot \cot 7^\circ) \cdot \tan 60^\circ \] Using the identity \(\tan\theta \cdot \cot\theta = 1\): \[ \text{LHS} = (1) \cdot \tan 60^\circ \] Now, substitute the known value of \(\tan 60^\circ\): \[ \text{LHS} = 1 \cdot \sqrt{3} = \sqrt{3} \] This is equal to the Right Hand Side (RHS).

Step 4: Final Answer:
Since LHS = RHS, the identity is proved.