{"id":715,"date":"2015-05-14T10:46:31","date_gmt":"2015-05-14T15:46:31","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=715"},"modified":"2015-05-14T10:46:31","modified_gmt":"2015-05-14T15:46:31","slug":"identity-from-math-stackexchange-com","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/identity-from-math-stackexchange-com\/","title":{"rendered":"Identity from Math.StackExchange.com"},"content":{"rendered":"\n<p>Taken from <a href=\"http:\/\/math.stackexchange.com\/a\/1281236\/409\">my answer<\/a> to <a href=\"http:\/\/math.stackexchange.com\/questions\/1281186\/proving-a-weird-trig-identity\/1281236#1281236\" title=\"&quot;Proving a weird trig identity&quot; at Math.StackExchange.com\">&#8220;Proving a weird trig identity&#8221;<\/a> at Math.StackExchange.com .<\/p>\n<hr \/>\n<p><\/p>\n<p><a href=\"http:\/\/daylateanddollarshort.com\/bloog\/wp-content\/uploads\/2015\/05\/mse-weirdidentity1.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/daylateanddollarshort.com\/bloog\/wp-content\/uploads\/2015\/05\/mse-weirdidentity1-1024x633.png\" alt=\"mse-weirdidentity\" width=\"480\" height=\"309\" class=\"aligncenter size-large wp-image-736\" \/><\/a><\/p>\n<p>$$\\Large{\\frac{\\cos\\theta}{1-\\sin\\theta} = \\frac{\\sec\\theta + \\tan\\theta}{1}}$$<\/p>\n<hr \/>\n<p><\/p>\n<p>In the diagram, \\(\\overline{AB}\\) is tangent to the unit circle at P. The &#8220;trig lengths&#8221; (except for \\(|\\overline{AQ}|\\)) should be clear.<\/p>\n<p>We note that \\(\\angle BPR \\cong \\angle RPP^\\prime\\), since these inscribed angles subtend congruent arcs \\(\\stackrel{\\frown}{PR}\\) and \\(\\stackrel{\\frown}{RP^\\prime}\\). Very little angle chasing gives that \\(\\triangle APQ\\) is isosceles, with \\(\\overline{AP}\\cong \\overline{AQ}\\) (justifying that last trig length). Then,<br \/>\n$$\\triangle SPR \\sim \\triangle OQR \\Longrightarrow \\frac{|\\overline{SP}|}{|\\overline{SR}|}=\\frac{|\\overline{OQ}|}{|\\overline{OR}|}\\Longrightarrow \\frac{\\cos\\theta}{1\u2212\\sin\\theta}=\\frac{\\sec\\theta+\\tan\\theta}{1}$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Taken from my answer to &#8220;Proving a weird trig identity&#8221; at Math.StackExchange.com . $$\\Large{\\frac{\\cos\\theta}{1-\\sin\\theta} = \\frac{\\sec\\theta + \\tan\\theta}{1}}$$ In the diagram, \\(\\overline{AB}\\) is tangent to the unit circle at P. The &#8220;trig lengths&#8221; (except for \\(|\\overline{AQ}|\\)) should be clear. We note that \\(\\angle BPR \\cong \\angle RPP^\\prime\\), since these inscribed angles subtend congruent arcs \\(\\stackrel{\\frown}{PR}\\) [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,5],"tags":[],"class_list":["post-715","post","type-post","status-publish","format-standard","hentry","category-classroom","category-trigonography"],"_links":{"self":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/715","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/comments?post=715"}],"version-history":[{"count":10,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/715\/revisions"}],"predecessor-version":[{"id":737,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/715\/revisions\/737"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=715"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=715"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=715"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}