Question Number 178992 by Spillover last updated on 23/Oct/22
Answered by som(math1967) last updated on 23/Oct/22
$$\theta_{\mathrm{2}} −\theta_{\mathrm{1}} =\theta_{\mathrm{3}} −\theta_{\mathrm{2}} =…=\theta_{{n}} −\theta_{{n}−\mathrm{1}} ={d} \\ $$$${sec}\theta_{\mathrm{1}} \mathrm{sec}\:\theta_{\mathrm{2}} +\mathrm{sec}\:\theta_{\mathrm{2}} \mathrm{sec}\:\theta_{\mathrm{1}} +…+{sec}\theta_{{n}−\mathrm{1}} {sec}\theta_{{n}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{cos}\:\theta_{\mathrm{1}} {cos}\theta_{\mathrm{2}} }\:+\frac{\mathrm{1}}{{cos}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{3}} }\:+…+\frac{\mathrm{1}}{{cos}\theta_{{n}−\mathrm{1}} {cos}\theta_{{n}} } \\ $$$$=\frac{\mathrm{1}}{{sind}}\left(\frac{{sind}}{\mathrm{cos}\:\theta_{\mathrm{1}} {cos}\theta_{\mathrm{2}} }\:+\frac{{sind}}{{cos}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{3}} }\:+…+\frac{{sind}}{{cos}\theta_{{n}−\mathrm{1}} {cos}\theta_{{n}} }\right) \\ $$$$=\frac{\mathrm{1}}{{sind}}\left\{\frac{{sin}\left(\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \right)}{\mathrm{cos}\:\theta_{\mathrm{1}} {cos}\theta_{\mathrm{2}} }\:+\frac{{sin}\left(\theta_{\mathrm{3}} −\theta_{\mathrm{2}} \right)}{{cos}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{3}} }\:+…+\frac{{sin}\left(\theta_{{n}} −\theta_{{n}−\mathrm{1}} \right)}{{cos}\theta_{{n}−\mathrm{1}} {cos}\theta_{{n}} }\right\} \\ $$$$=\frac{\mathrm{1}}{{sind}}\left\{\frac{{sin}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{1}} }{{cos}\theta_{\mathrm{1}} {cos}\theta_{\mathrm{2}} }\:−\frac{{sin}\theta_{\mathrm{1}} {cos}\theta_{\mathrm{2}} }{{cos}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{1}} }\right. \\ $$$$\:\:+\frac{{sin}\theta_{\mathrm{3}} {cos}\theta_{\mathrm{2}} }{{cos}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{3}} }−\frac{{sin}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{3}} }{{cos}\theta_{\mathrm{2}} {cos}\theta_{\mathrm{3}} }\:+… \\ $$$$\left.+\frac{{sin}\theta_{{n}} {cos}\theta_{{n}−\mathrm{1}} }{{cos}\theta_{{n}−\mathrm{1}} {cos}\theta_{{n}} }\:−\frac{{sin}\theta_{{n}−\mathrm{1}} {cos}\theta_{{n}} }{{cos}\theta_{{n}} {cos}\theta_{{n}−\mathrm{1}} }\right\} \\ $$$$=\frac{\mathrm{1}}{{sind}}\left\{{tan}\theta_{\mathrm{2}} −{tan}\theta_{\mathrm{1}} +{tan}\theta_{\mathrm{3}} −{tan}\theta_{\mathrm{2}} \right. \\ $$$$\left.+…+{tan}\theta_{{n}} −{tan}\theta_{{n}−\mathrm{1}} \right\} \\ $$$$=\frac{{tan}\theta_{{n}} −{tan}\theta_{\mathrm{1}} }{{sind}} \\ $$
Commented by Spillover last updated on 23/Oct/22
$$\mathrm{very}\:\mathrm{impressive}\:\mathrm{thank}\:\mathrm{you} \\ $$
Commented by Tawa11 last updated on 23/Oct/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$