Question

If $\cos (\alpha +\beta )\sin (\gamma +\delta )=\cos (\alpha -\beta )\sin (\gamma -\delta )$ then

• A. $\cot \alpha \cot \beta \cot \delta =\cot \gamma$
• B. $\cot \beta \cot \delta \cot \gamma =\cot \alpha$
• C. $\cot \alpha \cot \gamma \cot \delta =\cot \beta$
• D. $\cot \alpha \cot \beta \cot \gamma =\cot \delta$

Answer 1

The correct option is D $\cot \alpha \cot \beta \cot \gamma =\cot \delta$

Find $\cos (\alpha +\beta )\sin (r+\delta )=\cos (\alpha -\beta )\sin (r-\delta )$ then,

$\because \cos (a+b)=\cos a\cos b-\sin a\sin b$

$\cos (a-b)=\cos a\cos b+\sin a\sin b$

$\sin (a+b)=\sin a\cos b+\cos a\sin b$

$\sin (a-b)=\sin a\cos b-\cos a\sin b$

Now

$\cos (\alpha +\beta )\sin (r+\delta )=\cos (\alpha +\beta )\sin (r+\delta )$

$\Rightarrow \frac{\sin (r+\delta )}{\sin (r-\delta )}=\frac{\cos (\alpha +\beta )}{\cos \left(\alpha \beta \right)}$

$\Rightarrow \frac{\sin r\cos \delta +\cos r\sin \delta }{\sin r\cos \delta -\cos r\sin \delta }=\frac{\cos \alpha \cos \beta +\sin \alpha .\sin \beta }{\cos \alpha \cos \beta -\sin \alpha \sin \beta }$

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

$=\tan r.\cot \delta =\cot \alpha .\cot \beta$

$\Rightarrow \cot \alpha .\cot \beta .\cot r=\cot \delta$

$\therefore$ Option $D$ is correct