Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the general solution of the equation:

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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>



Answer
tanx space plus space tan open parentheses straight x plus straight pi over 3 close parentheses space plus space tan open parentheses straight x plus fraction numerator 2 straight pi over denominator 3 end fraction close parentheses space equals space 3
space space space space rightwards double arrow space space space space tanx space plus space fraction numerator tanx space plus space tan begin display style straight pi over 3 end style over denominator 1 minus tanx space tan begin display style straight pi over 3 end style end fraction plus fraction numerator tanx space plus space tan begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style over denominator 1 minus tanx space tan begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style end fraction space equals space 3
 rightwards double arrow   tanx space plus space fraction numerator tanx plus square root of 3 over denominator 1 minus square root of 3 space tanx end fraction plus fraction numerator tanx minus square root of 3 over denominator 1 plus square root of 3 tanx end fraction equals 3
 rightwards double arrow space tanx plus fraction numerator tanx plus square root of 3 over denominator 1 minus square root of 3 tanx end fraction plus fraction numerator tanx minus square root of 3 over denominator 1 plus square root of 3 space tanx end fraction space equals space 3

rightwards double arrow   tanx space plus fraction numerator left parenthesis tanx plus square root of 3 right parenthesis space left parenthesis 1 plus square root of 3 tanx right parenthesis space plus space left parenthesis tanx space minus square root of 3 right parenthesis space left parenthesis 1 minus square root of 3 space tanx right parenthesis over denominator left parenthesis 1 minus square root of 3 space tanx right parenthesis space left parenthesis 1 plus square root of 3 space tanx right parenthesis end fraction space equals space 3
rightwards double arrow   tanx space plus space fraction numerator tanx plus square root of 3 tan squared straight x plus square root of 3 plus 3 tanx space plus space tanx space minus space square root of 3 space tan squared straight x space minus space square root of 3 space plus space 3 tanx over denominator 1 minus 3 tan squared straight x end fraction space equals space 3

rightwards double arrow space space space tanx plus fraction numerator 8 tanx over denominator 1 minus 3 tan squared straight x end fraction equals 3           rightwards double arrow           fraction numerator tanx minus 3 tan cubed straight x plus 8 tanx over denominator 1 minus 3 tan squared straight x end fraction space equals space 3

rightwards double arrow space space space space fraction numerator 9 space tanx space minus space 3 tan cubed straight x over denominator 1 minus 3 tan squared straight x end fraction space equals space 3              rightwards double arrow    3 open parentheses fraction numerator 3 tanx space minus space tan cubed straight x over denominator 1 minus 3 tan squared straight x end fraction close parentheses space equals 3

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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>                            rightwards double arrow            tan 3 straight x space equals space tan straight pi over 4

rightwards double arrow       3 straight x space equals space nπ space plus space straight pi over 4 comma space space straight n space element of space straight Z           rightwards double arrow       straight x space equals space nπ over 3 plus straight pi over 12 comma space straight n space element of space straight Z


   

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