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$If \theta \textsubscript{1}, \theta \textsubscript{2}, \theta \textsubscript{3}, \ldots , \theta \textsubscript{n}$ are in A.P., whose common difference is d, show that $\\ \sec \theta_{1} \sec \theta_{2}+\sec \theta_{2} \sec \theta_{3}+\ldots+\sec \theta_{\mathrm{n}-1} \sec \theta_{\mathrm{n}} \\ = \frac{\tan \theta_{\mathrm{n}}-\tan \theta_{1}}{\sin \mathrm{d}}$

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Given that $\theta _{1}, \theta _{2}, \ldots \theta _{n}\text{~are in A.P and common difference is d. } \\\\$

$\\ \sec \theta _{1}\sec \theta _{2}+\sec \theta _{2}\sec \theta _{3}+ \ldots +\sec \theta _{n-1}\sec \theta _{n} \\\\ =\frac{1}{\cos \theta _{1}\cos \theta _{2}}+\frac{1}{\cos \theta _{2}\cos \theta _{3}}+ \ldots +\frac{1}{\cos \theta _{n-1}\cos \theta _{n}} \\\\ =\frac{\sin d}{\sin d} \left( \frac{1}{\cos \theta _{1}\cos \theta _{2}}+\frac{1}{\cos \theta _{2}\cos \theta _{3}}+ \ldots +\frac{1}{\cos \theta _{n-1}\cos \theta _{n}} \right) \\\\$
$\\ =\frac{1}{\sin d} \left( \frac{\sin \left( \theta _{2}- \theta _{1} \right) }{\cos \theta _{1}\cos \theta _{2}}+\frac{\sin \left( \theta _{3}- \theta _{2} \right) }{\cos \theta _{2}\cos \theta _{3}}+ \ldots +\frac{\sin \left( \theta _{n}- \theta _{n-1} \right) }{\cos \theta _{n-1}\cos \theta _{n}} \right) \\\\ =\frac{1}{\sin d} \left( \frac{\sin \theta _{2}\cos \theta _{1}-\cos \theta _{2}\sin \theta _{1}}{\cos \theta _{1}\cos \theta _{2}}+\frac{\sin \theta _{3}\cos \theta _{2}-\cos \theta _{3}\sin \theta _{2}}{\cos \theta _{2}\cos \theta _{3}}+ \ldots +\frac{\sin \theta _{n}\cos \theta _{n-1}-\cos \theta _{n}\sin \theta _{n-1}}{\cos \theta _{n-1}\cos \theta _{n}} \right) \\\\$
$\\=\frac{1}{\sin d} \left( \frac{\sin \theta _{2}\cos \theta _{1}}{\cos \theta _{1}\cos \theta _{2}}-\frac{\sin \theta _{1}\cos \theta _{2}}{\cos \theta _{1}\cos \theta _{2}}+\frac{\sin \theta _{3}\cos \theta _{2}}{\cos \theta _{2}\cos \theta _{3}}-\frac{\sin \theta _{2}\cos \theta _{3}}{\cos \theta _{2}\cos \theta _{3}}+ \ldots +\frac{\sin \theta _{n}\cos \theta _{n-1}}{\cos \theta _{n-1}\cos \theta _{n}}-\frac{\sin \theta _{n}\cos \theta _{n-1}}{\cos \theta _{n-1}\cos \theta _{n}} \right) \\\\ =\frac{1}{\sin d} \left( \tan \theta _{2}-\tan \theta _{1}+\tan \theta _{3}-\tan \theta _{2}+ \ldots +\tan \theta _{n}-\tan \theta _{n-1} \right) \\\\$

$\\ =\frac{ \left( \tan \theta _{n}-\tan \theta _{1} \right) }{\sin d} \\\\$

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