Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove that :

cot squared open parentheses straight pi over 6 close parentheses plus cosec open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses plus 3 tan squared open parentheses fraction numerator 7 straight pi over denominator 6 end fraction close parentheses space equals space 6

Answer

L.H.S. = cot squared open parentheses straight pi over 6 close parentheses plus cosec open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses plus 3 tan squared open parentheses fraction numerator 7 straight pi over denominator 6 end fraction close parentheses
space space space cot squared open parentheses straight pi over 6 close parentheses space equals space cot squared left parenthesis 30 degree right parenthesis space equals space left parenthesis square root of 3 right parenthesis squared space equals space 3
cosec open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses space equals cosec open parentheses fraction numerator 5 cross times 180 degree over denominator 6 end fraction close parentheses space equals space cosec 150 degree space equals space cosec left parenthesis 180 degree minus 30 degree right parenthesis
                   equals space cosec open parentheses straight pi minus straight pi over 6 close parentheses
cosec 150 degree is positive in quadrant II.
∴             <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>
                     tan open parentheses fraction numerator 7 straight pi over denominator 6 end fraction close parentheses space equals space tan 210 degree space equals space tan left parenthesis 180 degree plus 30 degree right parenthesis space equals space tan open parentheses straight pi plus straight pi over 6 close parentheses
tan 210 degree is +ve in quadrant III.
∴       tan open parentheses fraction numerator 7 straight pi over denominator 6 end fraction close parentheses space equals space tan open parentheses straight pi over 6 close parentheses space equals space fraction numerator 1 over denominator square root of 3 end fraction
         L.H.S. = cot squared open parentheses straight pi over 6 close parentheses plus cosec open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses plus 3 tan squared open parentheses fraction numerator 7 straight pi over denominator 6 end fraction close parentheses
                  equals 3 plus 2 plus 3 cross times open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses squared space equals space 3 plus 2 plus 1 space equals 6 space equals space straight R. straight H. straight S.
                   

More Chapters from Principle of Mathematical Induction