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Question
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
Options
- \[\frac{1 + k}{1 - k}\]
- \[\frac{1 - k}{1 + k}\]
- \[\frac{k + 1}{k - 1}\]
- \[\frac{k - 1}{k + 1}\]
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Solution
\[\frac{\cos( \theta_1 - \theta_2 )}{\cos( \theta_1 + \theta_2 )}\]
\[ = \frac{\cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2}{\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2}\]
\[\text{ Dividing numerator and denominator by }\cos \theta_1 \cos \theta_2 ,\text{ we get }:\]
\[\frac{1 + \tan \theta_1 \tan \theta_2}{1 - \tan \theta_1 \tan \theta_2}\]
\[ = \frac{1 + k}{1 - k}\]
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