Question

Tan1.tan2.tan3.........tan90

Answer 1

$\sf We\: know \:that\: tan\alpha= cot (90°-\alpha)\\</p><p>\sf Now, In\:the \:series\\</p><p>\sf tan1.tan2.tan3...tan44. tan45.tan46.tan47.....tan90\\</p><p>\sf tan45=1\\</p><p>\sf And \:after\: tan44,\: all \:the\: tan\alpha\: values\: can \:be\\\sf\: written\: in\: the \:form\: of\: cot(90°-\alpha)\\</p><p>\sf So \:now,\: the \:series \:appears \:like, \\\sf</p><p>tan1.tan2.tan3...tan44.cot44.cot43.....cot3cot2cot1\\</p><p>\sf Now, \:tan \:and\: cot\: will\: cancel\: out\: to \:1 \\</p><p>\sf Tan1.tan2.tan3.........tan90=1\\</p><p></p><p>\sf The\: answer\: is\: 1</p><p>$