Question

If \( \tan \text{A} = \frac{1}{\sqrt{x(x^2+x+1) \), \( \tan \text{B} = \frac{\sqrt{x}}{\sqrt{x^2+x+1}} \) and \( \tan \text{C} = \sqrt{x^{-1} + x^{-2} + x^{-3}} \) then}

• A. \( \text{A} + \text{B} = \text{C} \)
• B. \( A + B = 2C \)
• C. \( A + B = 3C \)
• D. \( A + B = 4C \)

Hint:

Look for common terms like \( \sqrt{x^2+x+1} \) to simplify complex algebraic trigonometric expressions.

Answer 1

The Correct Option is A

Step 1: Concept Use the trigonometric identity \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).

Step 2: Meaning Substitute the given values of \( \tan A \) and \( \tan B \) into the identity.

Step 3: Analysis
Numerator: \( \frac{1 + \sqrt{x} \cdot \sqrt{x}}{\sqrt{x(x^2+x+1)}} = \frac{1+x}{\sqrt{x(x^2+x+1)}} \).
Denominator: \( 1 - \frac{\sqrt{x}}{x(x^2+x+1)} = \frac{x^3+x^2+x-1}{x^3+x^2+x} \).
Simplifying \( \tan(A+B) \) and comparing with \( \tan C = \sqrt{\frac{x^2+x+1}{x^3}} \).


Step 4: Conclusion After simplification, we find \( \tan(A+B) = \tan C \), hence \( A + B = C \). Final Answer: (A)