Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove that:

2 space cos straight pi over 13 space cos fraction numerator 9 straight pi over denominator 13 end fraction plus cos fraction numerator 3 straight pi over denominator 13 end fraction plus cos fraction numerator 5 straight pi over denominator 13 end fraction space equals space 0

Answer

L.H.S. = 2 space cos open parentheses straight pi over 13 close parentheses space cos open parentheses fraction numerator 9 straight pi over denominator 13 end fraction close parentheses space plus space cos open parentheses fraction numerator 3 straight pi over denominator 13 end fraction close parentheses space plus space cos open parentheses fraction numerator 5 straight pi over denominator 13 end fraction close parentheses
          = cos open parentheses fraction numerator 9 straight pi over denominator 13 end fraction plus straight pi over 13 close parentheses space plus space cos open parentheses fraction numerator 9 straight pi over denominator 13 end fraction minus straight pi over 13 close parentheses plus cos open parentheses fraction numerator 3 straight pi over denominator 13 end fraction close parentheses plus cos open parentheses fraction numerator 5 straight pi over denominator 13 end fraction close parentheses
          =space space space cos open parentheses fraction numerator 10 straight pi over denominator 13 end fraction close parentheses plus cos open parentheses fraction numerator 8 straight pi over denominator 13 end fraction close parentheses plus cos open parentheses fraction numerator 3 straight pi over denominator 13 end fraction close parentheses plus cos open parentheses fraction numerator 5 straight pi over denominator 13 end fraction close parentheses
          = cos open parentheses straight pi minus fraction numerator 3 straight pi over denominator 13 end fraction close parentheses plus cos open parentheses straight pi minus fraction numerator 5 straight pi over denominator 13 end fraction close parentheses plus cos open parentheses fraction numerator 3 straight pi over denominator 13 end fraction close parentheses plus cos open parentheses fraction numerator 5 straight pi over denominator 13 end fraction close parentheses
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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>  left square bracket because space space cos space left parenthesis straight pi minus straight theta right parenthesis space equals space minus cosθ right square bracket
           = 0 = R.H.S.

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