Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Show that : <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#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
fraction numerator cos open parentheses begin display style fraction numerator 3 straight pi over denominator 4 end fraction end style close parentheses minus cos open parentheses begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style close parentheses over denominator cos open parentheses begin display style fraction numerator 3 straight pi over denominator 4 end fraction end style close parentheses plus cos open parentheses begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style close parentheses end fraction space equals space fraction numerator cos 135 degree minus cos 120 degree over denominator cos 135 degree plus cos 120 degree end fraction
Now,  space space cos 135 degree space equals space cos left parenthesis 180 degree minus 45 degree right parenthesis space equals space cos open parentheses straight pi minus straight pi over 4 close parentheses space equals space minus cos straight pi over 4 space equals space minus fraction numerator 1 over denominator square root of 2 end fraction

           cos open parentheses fraction numerator 2 straight pi over denominator 3 end fraction close parentheses space equals space cos space 120 degree space equals space cos left parenthesis 180 degree minus 60 degree right parenthesis space equals space cos open parentheses straight pi minus straight pi over 3 close parentheses space equals space minus cos open parentheses straight pi over 3 close parentheses space equals space minus 1 half
∴   fraction numerator cos open parentheses begin display style fraction numerator 3 straight pi over denominator 4 end fraction end style close parentheses minus cos open parentheses begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style close parentheses over denominator cos open parentheses begin display style fraction numerator 3 straight pi over denominator 4 end fraction end style close parentheses plus cos open parentheses begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style close parentheses end fraction space equals space fraction numerator negative begin display style fraction numerator 1 over denominator square root of 2 end fraction end style plus begin display style 1 half end style over denominator negative begin display style fraction numerator 1 over denominator square root of 2 end fraction end style minus begin display style 1 half end style end fraction space equals space fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style minus begin display style 1 half end style over denominator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style plus begin display style 1 half end style end fraction space equals space fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction space equals space fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction cross times fraction numerator square root of 2 minus 1 over denominator square root of 2 minus 1 end fraction

                                       equals space space fraction numerator 2 plus 1 minus 2 square root of 2 over denominator 2 minus 1 end fraction space equals space 3 minus 2 square root of 2
Hence,         <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#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>

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