![\"\"](\"https:\/\/mdr.foobrdigital.com\/wp-content\/uploads\/2022\/09\/qqqq.png\")
<\/figure>\n\n\n\nNow, we can prove these results using trigonometric formulas.<\/p>\n\n\n\n
Prove that for any real numbers x and y, sin x = sin y implies x = n\u03c0 + (-1)n<\/sup>y, where n \u2208 Z<\/h3>\n\n\n\nProof: <\/strong>If sin x = sin y, then sin x \u2013 sin y = 0<\/p>\n\n\n\n\u21d2 2 cos (x + y)\/2 sin (x \u2212 y)\/2 = 0 — [Using formula\u00a0Sin A – Sin B = 2 cos \u00bd (A + B) sin \u00bd (A – B)]<\/p>\n\n\n\n
\u21d2 cos (x + y)\/2 = 0 or sin (x \u2212 y)\/2 = 0<\/p>\n\n\n\n
\u21d2 (x + y)\/2 = (2n + 1)\u03c0 \/2 or (x \u2212 y)\/2 = n\u03c0, where n \u2208 Z —- [Because sin A = 0 implies A = n\u03c0 and cos A = 0 implies A = (2n + 1)\u03c0\/2, where n \u2208 Z]<\/p>\n\n\n\n
i.e. x = (2n + 1) \u03c0 \u2013 y or x = 2n\u03c0 + y, where n \u2208 Z.<\/p>\n\n\n\n
Hence x = (2n + 1)\u03c0 + (\u20131)2n + 1<\/sup>y or x = 2n\u03c0 + (\u20131)2n<\/sup> y, where n \u2208 Z.<\/p>\n\n\n\nCombining these two results, we get x = n\u03c0 + (\u20131)n<\/sup>y, where n \u2208 Z.<\/p>\n\n\n\nProve that for any real numbers x and y, cos x = cos y implies x = 2n\u03c0 \u00b1 y, where n \u2208 Z.<\/h3>\n\n\n\n
Proof: <\/strong>If cos x = cos y, then cos x \u2013 cos y = 0<\/p>\n\n\n\n\u21d2 -2 sin (x + y)\/2 sin (x \u2212 y)\/2 = 0 — [Using formula\u00a0Cos A – Cos B = – 2 sin \u00bd (A + B) sin \u00bd (A – B)]<\/p>\n\n\n\n
\u21d2 sin (x + y)\/2 = 0 or sin (x \u2212 y)\/2 = 0<\/p>\n\n\n\n
\u21d2 (x + y)\/2 = n\u03c0 or (x \u2212 y)\/2 = n\u03c0, where n \u2208 Z —- [Because sin A = 0 implies A = n\u03c0, where n \u2208 Z]<\/p>\n\n\n\n
i.e. x = 2n\u03c0 \u2013 y or x = 2n\u03c0 + y, where n \u2208 Z.<\/p>\n\n\n\n
Hence x = 2n\u03c0 \u00b1 y, where n \u2208 Z.<\/p>\n\n\n\n
Prove that if x and y are not odd multiples of \u03c0\/2, then tan x = tan y implies x = n\u03c0 + y, where n \u2208 Z.<\/h3>\n\n\n\n
Proof: <\/strong>If tan x = tan y, then tan x – tan y = 0<\/p>\n\n\n\n\u21d2 sin x \/ cos x – sin y \/ cos y = 0<\/p>\n\n\n\n
\u21d2 (sin x cos y – cos x sin y) \/ (cos x cos y) = 0<\/p>\n\n\n\n
\u21d2 sin (x – y) \/ (cos x cos y) = 0 —- <\/p>\n\n\n\n
[Using trigonometric formula\u00a0sin (A – B) = sinA cosB – sinB cosA]<\/p>\n\n\n\n
\u21d2 sin (x – y) = 0<\/p>\n\n\n\n
\u21d2 x – y = n\u03c0, where n \u2208 Z — [Because sin A = 0 implies A = n\u03c0, where n \u2208 Z]<\/p>\n\n\n\n
\u21d2 x = n\u03c0 + y, where n \u2208 Z<\/p>\n","protected":false},"excerpt":{"rendered":"
We use some results and general solutions of the basic trigonometric equations to solve other trigonometric equations. These results are as follows: For any real numbers x and y, sin x = sin y implies x = n\u03c0 + (-1)ny, where n \u2208 Z. For any real numbers x and y, cos x = cos […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[384],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7765"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=7765"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7765\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=7765"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=7765"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=7765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u21d2 2 cos (x + y)\/2 sin (x \u2212 y)\/2 = 0 — [Using formula\u00a0Sin A – Sin B = 2 cos \u00bd (A + B) sin \u00bd (A – B)]<\/p>\n\n\n\n
\u21d2 cos (x + y)\/2 = 0 or sin (x \u2212 y)\/2 = 0<\/p>\n\n\n\n
\u21d2 (x + y)\/2 = (2n + 1)\u03c0 \/2 or (x \u2212 y)\/2 = n\u03c0, where n \u2208 Z —- [Because sin A = 0 implies A = n\u03c0 and cos A = 0 implies A = (2n + 1)\u03c0\/2, where n \u2208 Z]<\/p>\n\n\n\n
i.e. x = (2n + 1) \u03c0 \u2013 y or x = 2n\u03c0 + y, where n \u2208 Z.<\/p>\n\n\n\n
Hence x = (2n + 1)\u03c0 + (\u20131)2n + 1<\/sup>y or x = 2n\u03c0 + (\u20131)2n<\/sup> y, where n \u2208 Z.<\/p>\n\n\n\n Combining these two results, we get x = n\u03c0 + (\u20131)n<\/sup>y, where n \u2208 Z.<\/p>\n\n\n\n Proof: <\/strong>If cos x = cos y, then cos x \u2013 cos y = 0<\/p>\n\n\n\n \u21d2 -2 sin (x + y)\/2 sin (x \u2212 y)\/2 = 0 — [Using formula\u00a0Cos A – Cos B = – 2 sin \u00bd (A + B) sin \u00bd (A – B)]<\/p>\n\n\n\n \u21d2 sin (x + y)\/2 = 0 or sin (x \u2212 y)\/2 = 0<\/p>\n\n\n\n \u21d2 (x + y)\/2 = n\u03c0 or (x \u2212 y)\/2 = n\u03c0, where n \u2208 Z —- [Because sin A = 0 implies A = n\u03c0, where n \u2208 Z]<\/p>\n\n\n\n i.e. x = 2n\u03c0 \u2013 y or x = 2n\u03c0 + y, where n \u2208 Z.<\/p>\n\n\n\n Hence x = 2n\u03c0 \u00b1 y, where n \u2208 Z.<\/p>\n\n\n\n Proof: <\/strong>If tan x = tan y, then tan x – tan y = 0<\/p>\n\n\n\n \u21d2 sin x \/ cos x – sin y \/ cos y = 0<\/p>\n\n\n\n \u21d2 (sin x cos y – cos x sin y) \/ (cos x cos y) = 0<\/p>\n\n\n\n \u21d2 sin (x – y) \/ (cos x cos y) = 0 —- <\/p>\n\n\n\n [Using trigonometric formula\u00a0sin (A – B) = sinA cosB – sinB cosA]<\/p>\n\n\n\n \u21d2 sin (x – y) = 0<\/p>\n\n\n\n \u21d2 x – y = n\u03c0, where n \u2208 Z — [Because sin A = 0 implies A = n\u03c0, where n \u2208 Z]<\/p>\n\n\n\n \u21d2 x = n\u03c0 + y, where n \u2208 Z<\/p>\n","protected":false},"excerpt":{"rendered":" We use some results and general solutions of the basic trigonometric equations to solve other trigonometric equations. These results are as follows: For any real numbers x and y, sin x = sin y implies x = n\u03c0 + (-1)ny, where n \u2208 Z. For any real numbers x and y, cos x = cos […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[384],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7765"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=7765"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7765\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=7765"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=7765"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=7765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Prove that for any real numbers x and y, cos x = cos y implies x = 2n\u03c0 \u00b1 y, where n \u2208 Z.<\/h3>\n\n\n\n
Prove that if x and y are not odd multiples of \u03c0\/2, then tan x = tan y implies x = n\u03c0 + y, where n \u2208 Z.<\/h3>\n\n\n\n