Question

The smallest positive value (in degrees) of \( \theta \) for which \( \tan(\theta + 100^\circ) = \tan(\theta + 50^\circ) \tan(\theta - 50^\circ) \) is valid, is:

• A. \( 60^\circ \)
• B. \( 45^\circ \)
• C. \( 30^\circ \)
• D. \( 15^\circ \)

Hint:

For trigonometric equations, use identities to simplify the equation and solve for \( \theta \).

Answer 1

The Correct Option is C

We are given the equation: \[ \tan(\theta + 100^\circ) = \tan(\theta + 50^\circ) \tan(\theta - 50^\circ) \] 

Step 1: Recall Trigonometric Identity 
Using the identity: \[ \tan A \tan B = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} \] We'll simplify the right side using this identity. 

Step 2: Identifying the Values 
From the given equation: \[ \tan(\theta + 100^\circ) = \tan(\theta + 50^\circ) \tan(\theta - 50^\circ) \] 

Step 3: Use Identity for Product of Tangents 
Using the identity for tangent product, \[ \tan(A) \tan(B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} \] Substituting the known angles, \[ \tan(\theta + 100^\circ) = \frac{\tan(\theta + 50^\circ) + \tan(\theta - 50^\circ)}{1 - \tan(\theta + 50^\circ)\tan(\theta - 50^\circ)} \] 

Step 4: Solving for \( \theta \) 
By simplifying both sides and using the tangent addition and subtraction identities, the equation simplifies to: \[ \theta = 30^\circ \] 

Final Answer: (3) \( 30^\circ \)