Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove 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

L.H.S. = cos 2 straight x space cos straight x over 2 space minus space cos 3 straight x space cos fraction numerator 9 straight x over denominator 2 end fraction space equals space cos 2 straight x space cos straight x over 2 minus cos fraction numerator 9 straight x over denominator 2 end fraction cos 3 straight x
         = 1 half open square brackets 2 cos 2 straight x space cos straight x over 2 minus space 2 cos fraction numerator 9 straight x over denominator 2 end fraction cos 3 straight x close square brackets
         =space space 1 half open curly brackets cos open parentheses 2 straight x plus straight x over 2 close parentheses plus cos open parentheses 2 straight x minus straight x over 2 close parentheses close curly brackets minus open curly brackets cos open parentheses fraction numerator 9 straight x over denominator 2 end fraction plus 3 straight x close parentheses plus cos open parentheses fraction numerator 9 straight x over denominator 2 end fraction minus 3 straight x close parentheses close curly brackets
          = 1 half open square brackets cos fraction numerator 5 straight x over denominator 2 end fraction plus cos fraction numerator 3 straight x over denominator 2 end fraction minus cos fraction numerator 15 straight x over denominator 2 end fraction minus cos fraction numerator 3 straight x over denominator 2 end fraction close square brackets
          = 1 half open square brackets cos fraction numerator 5 straight x over denominator 2 end fraction minus cos fraction numerator 15 straight x over denominator 2 end fraction close square brackets =space space minus 1 half open square brackets cos fraction numerator 15 straight x over denominator 2 end fraction minus cos fraction numerator 5 straight x over denominator 2 end fraction close square brackets
          equals fraction numerator negative 1 over denominator 2 end fraction open square brackets negative 2 sin open parentheses fraction numerator begin display style fraction numerator 15 straight x over denominator 2 end fraction end style plus begin display style fraction numerator 5 straight x over denominator 2 end fraction end style over denominator 2 end fraction close parentheses sin open parentheses fraction numerator begin display style fraction numerator 15 straight x over denominator 2 end fraction end style minus begin display style fraction numerator 5 straight x over denominator 2 end fraction end style over denominator 2 end fraction close parentheses close square brackets
            = sin 5 straight x space sin open parentheses fraction numerator 5 straight x over denominator 2 end fraction close parentheses space equals space straight R. straight H. straight S.
Hence,         L.H.S. = R.H.S.

More Chapters from Principle of Mathematical Induction