Question

Assertion :${\left(\frac{\cos A+\cos B}{\sin A-\sin B}\right)}^n+{\left(\frac{\sin A+\sin B}{\cos A-\cos B}\right)}^n$
$=2{\cot }^n\left(\frac{A-B}{2}\right)$ if $n$ is even
$=0$ if $n$ is odd. Reason: $\frac{\cos A+\cos B}{\sin A-\sin B}=\cot \frac{A-B}{2}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\frac{\cos A+\cos B}{\sin A-\sin B}=\frac{2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)}{2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)}=\cot \left(\frac{A-B}{2}\right)$

$\frac{\sin A+\sin B}{\cos A-\cos B}=\frac{2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)}{-2\sin \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)}=-\cot \left(\frac{A-B}{2}\right)$

${\left(\frac{\cos A+\cos B}{\sin A-\sin B}\right)}^n+{\left(\frac{\sin A+\sin B}{\cos A-\cos B}\right)}^n={\left(\cot \left(\frac{A-B}{2}\right)\right)}^n+{(-\cot \left(\frac{A-B}{2}\right))}^n$

If $n$ is even
${\left(\frac{\cos A+\cos B}{\sin A-\sin B}\right)}^n+{\left(\frac{\sin A+\sin B}{\cos A-\cos B}\right)}^n=2\cot \left(\frac{A-B}{2}\right)$
Hence, option 'A' is correct.