Question

The expression $\frac{\sin (\alpha +\theta )-\sin (\alpha -\theta )}{\cos (\beta -\theta )-\cos (\beta +\theta )}$ is

• A. independent of $\alpha$
• B. independent of $\beta$
• C. independent of $\theta$
• D. independent of $\alpha$ and $\beta$

Answer 1

Answer: (C)

independent of $\theta$

Given $\frac{\sin (\alpha +\theta )-\sin (\alpha -\theta )}{\cos (\beta -\theta )-\cos (\beta +\theta )}$

Simplifying, we get

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

$=\frac{\sin \left(\theta \right)\cos \left(\alpha \right)}{\sin \left(\theta \right)\sin \left(\beta \right)}$

$=\frac{\cos \alpha }{\sin \beta }$

Hence independent of $\theta$.