Question

If \( \tan \left( \frac{\alpha + \beta}{2} \right) \), \( \cos (\alpha + \beta) \), \( \sin (\alpha + \beta) \), and \( \tan (\alpha + \beta) \) are matched with their values, then the correct matching is:

• A. \( (I) \rightarrow (a) \), \( (II) \rightarrow (e) \), \( (III)\rightarrow (b)\)
• B. \( (I) \rightarrow (a) \), \( (II) \rightarrow (c) \), \( (III) \rightarrow (b) \), \( (IV) \rightarrow (e) \)
• C. \( (I) \rightarrow (a) \), \( (II) \rightarrow (d) \), \( (III) \rightarrow (c) \), \( (IV) \rightarrow (b) \)
• D. \( (I) \rightarrow (a) \), \( (II) \rightarrow (d) \), \( (III) \rightarrow (b) \), \( (IV) \rightarrow (c) \)

Hint:

When matching trigonometric identities, refer to the sum formulas and use the standard trigonometric identities for addition of angles.

Answer 1

The Correct Option is D

Step 1: Apply sum-to-product identities Using the sum-to-product identities for \( \cos(\alpha + \beta) \) and \( \sin(\alpha + \beta) \): - \( \tan \left( \frac{\alpha + \beta}{2} \right) = \frac{b}{a} \) because it is derived from the sum and difference identities for sine and cosine. Step 2: Use trigonometric sum formulas - \( \cos(\alpha + \beta) \) and \( \sin(\alpha + \beta) \) simplify to the expressions: \[ \cos(\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2} \] \[ \sin(\alpha + \beta) = \frac{2ab}{a^2 + b^2} \] - \( \tan(\alpha + \beta) \) simplifies to: \[ \tan(\alpha + \beta) = \frac{2ab}{a^2 - b^2} \] Step 3: Match the items with List B Now, we match the results from List - A with the corresponding expressions in List - B:
- \( (I) \rightarrow (a) \): \( \tan \left( \frac{\alpha + \beta}{2} \right) = \frac{b}{a} \)
- \( (II) \rightarrow (d) \): \( \cos(\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2} \)
- \( (III) \rightarrow (b) \): \( \sin(\alpha + \beta) = \frac{2ab}{a^2 + b^2} \)
- \( (IV) \rightarrow (c) \): \( \tan(\alpha + \beta) = \frac{2ab}{a^2 - b^2} \)
Thus, the correct matching is option (4).