Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove that:

cosα space plus space cosβ space plus space cosγ space plus space cos space left parenthesis straight alpha space plus space straight beta space plus space straight gamma right parenthesis space equals space 4 cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator straight beta plus straight gamma over denominator 2 end fraction close parentheses space cos fraction numerator straight gamma plus straight alpha over denominator 2 end fraction.

Answer

L.H.S. = space space cosα space plus space cosβ space plus space cosγ space plus space cos space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis
         equals space left square bracket space cosα space plus space cosβ right square bracket space plus space left square bracket cos space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis space plus space cosγ right square bracket
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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>
          equals 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 plus space 2 space cos open parentheses fraction numerator straight alpha plus straight beta plus 2 straight gamma over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses  
           <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>
   equals space 2 space cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses space open square brackets cos open parentheses fraction numerator straight alpha plus straight beta plus 2 straight gamma over denominator 2 end fraction close parentheses plus cos open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses close square brackets
   equals space 2 space cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses open square brackets 2 space cos open parentheses fraction numerator begin display style fraction numerator straight alpha plus straight beta plus 2 straight gamma over denominator 2 end fraction end style plus begin display style fraction numerator straight alpha minus straight beta over denominator 2 end fraction end style over denominator 2 end fraction close parentheses close square brackets. space cos open parentheses fraction numerator begin display style fraction numerator straight alpha plus straight beta plus 2 straight gamma over denominator 2 end fraction end style minus begin display style fraction numerator straight alpha minus straight beta over denominator 2 end fraction end style over denominator 2 end fraction close parentheses
   equals space 4 space cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator 2 straight alpha plus 2 straight gamma over denominator 4 end fraction close parentheses space cos open parentheses fraction numerator 2 straight beta plus 2 straight gamma over denominator 4 end fraction close parentheses
    equals space 4 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 beta plus straight gamma over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator straight gamma plus straight alpha over denominator 2 end fraction close parentheses = R.H.S.
Hence,   L.H.S. = R.H.S.
   
   

        
          

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