Principle of Mathematical Induction

Principle of Mathematical Induction

Question

If          sin space straight alpha space plus space sin space straight beta space equals space straight a

           space space cos space straight alpha space plus space cos space straight beta space equals space straight b comma

show that:

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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

 sin space straight alpha space plus space sin space straight beta space equals space straight a   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>                ...(i)

space space cos space straight alpha space plus space cos space straight beta space equals space straight b rightwards double arrow  space space space 2 space cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses space equals straight b              ...(ii)
Dividing (i) and (ii), we get
fraction numerator 2 sin open parentheses begin display style fraction numerator straight alpha plus straight beta over denominator 2 end fraction end style close parentheses space cos open parentheses begin display style fraction numerator straight alpha minus straight beta over denominator 2 end fraction end style close parentheses over denominator 2 cos open parentheses begin display style fraction numerator straight alpha plus straight beta over denominator 2 end fraction end style close parentheses space cos open parentheses begin display style fraction numerator straight alpha minus straight beta over denominator 2 end fraction end style close parentheses end fraction space equals space straight a over straight b        rightwards double arrow          tan open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses space equals space straight a over straight b

cos left parenthesis straight alpha plus straight beta right parenthesis space equals space fraction numerator 1 minus tan squared open parentheses begin display style fraction numerator straight alpha plus straight beta over denominator 2 end fraction end style close parentheses over denominator 1 plus tan squared open parentheses begin display style fraction numerator straight alpha plus straight beta over denominator 2 end fraction end style close parentheses end fraction                       open parentheses because space cos 2 straight x space equals space fraction numerator 1 minus tan squared space straight x over denominator 1 plus tan squared space straight x end fraction close parentheses
                = fraction numerator 1 minus begin display style straight a squared over straight b squared end style over denominator 1 plus begin display style straight a squared over straight b squared end style end fraction space equals space fraction numerator straight b squared minus straight a squared over denominator straight b squared plus straight a squared end fraction
Hence,     cos left parenthesis straight alpha space plus space straight beta right parenthesis space equals space fraction numerator straight b squared minus straight a squared over denominator straight b squared plus straight a squared end fraction

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