Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove the following:

sin open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses plus cos open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses space equals space cos open parentheses straight pi over 4 close parentheses space equals fraction numerator 1 over denominator square root of 2 end fraction


Answer
sin open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses plus cos open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses space equals space sin 105 degree plus cos 105 degree space equals space sin 105 degree plus cos left parenthesis 90 degree plus 15 degree right parenthesis
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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>           [∵ cos left parenthesis 90 degree plus straight theta right parenthesis space equals space minus sinθ]
                               equals 2 cos open parentheses fraction numerator 105 degree plus 15 degree over denominator 2 end fraction close parentheses sin open parentheses fraction numerator 105 degree minus 15 degree over denominator 2 end fraction close parentheses
                                equals 2 space cos space 60 degree space sin 45 degree space equals space 2 open parentheses 1 half close parentheses space cross times space sin 45 degree space equals space sin open parentheses straight pi over 4 close parentheses space equals space cos open parentheses straight pi over 4 close parentheses space equals space fraction numerator 1 over denominator square root of 2 end fraction
Hence,    sin open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses plus cos open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses space equals space cos open parentheses straight pi over 4 close parentheses space equals space fraction numerator 1 over denominator square root of 2 end fraction
                                

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