Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Solve the equation:

secθ space minus space 1 space equals space left parenthesis square root of 2 minus 1 right parenthesis space tanθ  where straight theta not equal to left parenthesis 2 straight n plus 1 right parenthesis straight pi over 2 comma space straight n space element of space straight Z

Answer

                   secθ space minus 1 space equals space left parenthesis square root of 2 minus 1 right parenthesis space tanθ
rightwards double arrow               secθ minus 1 space equals space tan straight pi over 8 tanθ                            open square brackets because space tan straight pi over 8 space equals space square root of 2 minus 1 close square brackets
rightwards double arrow                1 over cosθ minus 1 space equals space fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction. space sinθ over cosθ
rightwards double arrow                 fraction numerator 1 minus cosθ over denominator cosθ end fraction space equals space fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction. space sinθ over cosθ     space space open square brackets because space space cosθ space not equal to 0 space as space 0 not equal to space left parenthesis 2 straight n plus 1 right parenthesis straight pi over 2 comma space straight n space element of space straight Z close square brackets
rightwards double arrow                 1 minus cosθ space equals space fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction. sinθ
rightwards double arrow           cos straight pi over 8 minus space cosθ. space cos straight pi over 8 space equals space sin straight pi over 8. space sinθ
rightwards double arrow                            cos straight pi over 8 space equals space cosθ. space cos straight pi over 8 plus sinθ. space sin straight pi over 8
rightwards double arrow                             space space cos straight pi over 8 space equals space cos open parentheses straight theta space minus space straight pi over 8 close parentheses
rightwards double arrow             <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>
rightwards double arrow                      space space straight theta minus straight pi over 8 space equals space 2 nπ plus-or-minus straight pi over 8 comma space space straight n space element of space straight Z
rightwards double arrow                         <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>
rightwards double arrow                          space space space straight theta space equals space 2 nπ space plus space straight pi over 4 space or space space straight theta space equals space 2 nπ comma space straight n space element of space straight Z

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