Question

If \(A, B, C\) are the angles of \(\triangle ABC\), then with usual notations, \[ \frac{c^2 - a^2 + b^2}{a^2 - b^2 + c^2} = \]

• A. \( \dfrac{\cos B}{\cos A} \)
• B. \( \dfrac{\cot B}{\cot A} \)
• C. \( \dfrac{\sin B}{\sin A} \)
• D. \( \dfrac{\tan B}{\tan A} \)

Hint:

Expressions involving sides squared often simplify using cosine rule identities.

Answer 1

The Correct Option is D

Step 1: Use the cosine rule.
Using cosine rule in \(\triangle ABC\): \[ c^2 = a^2 + b^2 - 2ab\cos C \]
Step 2: Simplify numerator and denominator.
\[ c^2 - a^2 + b^2 = 2b^2 - 2ab\cos C \] \[ a^2 - b^2 + c^2 = 2a^2 - 2ab\cos C \]
Step 3: Take ratio.
\[ \frac{c^2 - a^2 + b^2}{a^2 - b^2 + c^2} = \frac{b(b - a\cos C)}{a(a - b\cos C)} = \frac{\tan B}{\tan A} \]