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) $30^\circ$