# sin5-cos0-cos5-sin5-cos5-cos10-sin5-cos55-cos60-

Question Number 131230 by Fikret last updated on 02/Feb/21
$$\frac{{sin}\mathrm{5}}{{cos}\mathrm{0}.{cos}\mathrm{5}}+\frac{{sin}\mathrm{5}}{{cos}\mathrm{5}.{cos}\mathrm{10}}+\:\:.\:\:.\:\:.\:\:+\frac{{sin}\mathrm{5}}{{cos}\mathrm{55}.{cos}\mathrm{60}}\:=? \\$$
Answered by mr W last updated on 02/Feb/21
$${example}: \\$$$$\mathrm{sin}\:\mathrm{5}=\mathrm{sin}\:\left(\mathrm{10}−\mathrm{5}\right)=\mathrm{sin}\:\mathrm{10}\:\mathrm{cos}\:\mathrm{5}−\mathrm{cos}\:\mathrm{10}\:\mathrm{sin}\:\mathrm{5} \\$$$$\frac{\mathrm{sin}\:\mathrm{5}}{\mathrm{cos}\:\mathrm{5}\:\mathrm{cos}\:\mathrm{10}}=\mathrm{tan}\:\mathrm{10}−\mathrm{tan}\:\mathrm{5} \\$$$$\\$$$$\frac{{sin}\mathrm{5}}{{cos}\mathrm{0}.{cos}\mathrm{5}}+\frac{{sin}\mathrm{5}}{{cos}\mathrm{5}.{cos}\mathrm{10}}+\:\:.\:\:.\:\:.\:\:+\frac{{sin}\mathrm{5}}{{cos}\mathrm{55}.{cos}\mathrm{60}} \\$$$$=\left(\mathrm{tan}\:\mathrm{5}−\mathrm{tan}\:\mathrm{0}\right)+\left(\mathrm{tan}\:\mathrm{10}−\mathrm{tan}\:\mathrm{5}\right)+…+\left(\mathrm{tan}\:\mathrm{60}−\mathrm{tan}\:\mathrm{55}\right) \\$$$$=\mathrm{tan}\:\mathrm{60}−\mathrm{tan}\:\mathrm{0} \\$$$$=\mathrm{tan}\:\mathrm{60} \\$$$$=\sqrt{\mathrm{3}} \\$$
Commented by malwan last updated on 02/Feb/21
$${great}\:{sir} \\$$$${thank}\:{you}\: \\$$
Commented by PRITHWISH SEN 2 last updated on 03/Feb/21
$$\boldsymbol{\mathrm{Beautiful}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{always}}\:! \\$$