The expression (sin(alpha + theta) - sin(alpha - theta))/(cos(beta - theta) - cos(beta + theta)) * is - (A) independent of a (C) independent of (B) independent of B (D) independent of a and B

Key Identities to Know

1. Sine difference identity
   $\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$
2. Cosine difference identity
   $\cos C - \cos D = -2 \sin\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right)$

Applying the Identities Step-by-Step

1. Numerator
   Take $A = \alpha + \theta$ and $B = \alpha - \theta$

   $\sin(\alpha + \theta) - \sin(\alpha - \theta) = 2\,\cos\left(\frac{(\alpha+\theta)+(\alpha-\theta)}{2}\right) \sin\left(\frac{(\alpha+\theta)-(\alpha-\theta)}{2}\right)$

   Simplify inside the brackets: $\cos(\alpha)\,\sin(\theta)$ Multiplied by the 2 outside gives $2\cos(\alpha)\sin(\theta)$

2. Denominator
   Take $C = \beta - \theta$ and $D = \beta + \theta$

   $\cos(\beta - \theta) - \cos(\beta + \theta) = -2\,\sin\left(\frac{(\beta-\theta)+(\beta+\theta)}{2}\right) \sin\left(\frac{(\beta-\theta)-(\beta+\theta)}{2}\right)$

   First sine term becomes $\sin(\beta)$ and the second becomes $\sin(-\theta)= -\sin(\theta)$. The two minus signs cancel, giving $2\sin(\beta)\sin(\theta)$

3. Form the fraction
   $\frac{2\cos(\alpha)\sin(\theta)}{2\sin(\beta)\sin(\theta)} = \frac{\cos(\alpha)}{\sin(\beta)}$

Notice $\sin(\theta)$ cancels out completely. Therefore the final result depends only on $\alpha$ and $\beta$, not on $\theta$.

Conclusion

The expression is independent of $\theta$. In the given option-pattern, that corresponds to choice (C).

Imagine you have two spinning wheels. One wheel is marked with the angle alpha (α) and you can nudge it forward or backward by another small angle theta (θ). You do the same with a second wheel marked beta (β). When you plug those spun-wheel readings into the funny fraction

   sin(α + θ) – sin(α – θ)
--------------------------------
   cos(β – θ) – cos(β + θ)

all the extra twirls of θ on both wheels mysteriously cancel each other out! What you’re left with only depends on how the wheels were marked at the start (α and β), not on how much you nudged them (θ). So the value of the whole fraction does not care about θ at all.

Step-by-Step Solution

1. Numerator

   $\sin(\alpha + \theta) - \sin(\alpha - \theta) = 2\cos\left(\frac{(\alpha+\theta)+(\alpha-\theta)}{2}\right) \sin\left(\frac{(\alpha+\theta)-(\alpha-\theta)}{2}\right)$

   $= 2\cos(\alpha)\sin(\theta)$

2. Denominator

   $\cos(\beta - \theta) - \cos(\beta + \theta) = -2\sin\left(\frac{(\beta-\theta)+(\beta+\theta)}{2}\right) \sin\left(\frac{(\beta-\theta)-(\beta+\theta)}{2}\right)$

   $= -2\sin(\beta)\sin(-\theta)$

   $= 2\sin(\beta)\sin(\theta)$

3. Form the ratio

   $\frac{2\cos(\alpha)\sin(\theta)}{2\sin(\beta)\sin(\theta)} = \frac{\cancel{2}\;\cos(\alpha)\;\cancel{\sin(\theta)}}{\cancel{2}\;\sin(\beta)\;\cancel{\sin(\theta)}} = \frac{\cos\alpha}{\sin\beta}$

4. Observation

   The final answer contains no $\theta$. Hence the value is independent of $\theta$.

Answer: (C) independent of $\theta$