Principle Of Mathematical Induction

Question
CBSEENMA11014336

Deduce cosine formula from sine formula

Solution

We have to deduce cosine formula from sine formula i.e. we have to show
cos space straight A space equals space fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction
R.H.S. = fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction
          equals fraction numerator straight k squared sin squared straight B plus straight k squared sin squared straight C minus straight k squared sin squared straight A over denominator 2 straight k squared sinBsinC end fraction
           equals fraction numerator sin squared straight B space plus space sin squared straight C space minus space sin squared straight A over denominator 2 space sinB space sinC end fraction            [By sine formula]
           <pre>uncaught exception: <b>file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/5d/7e/af0987dcc76cef9981594f555ff6.ini): failed to open stream: Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php at line #12file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/5d/7e/af0987dcc76cef9981594f555ff6.ini): failed to open stream: Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php line 12<br />#0 [internal function]: _hx_error_handler(2, 'file_put_conten...', '/home/config_ad...', 12, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php(12): file_put_contents('/home/config_ad...', 'mml=<math xmlns...')
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(48): sys_io_File::saveContent('/home/config_ad...', 'mml=<math xmlns...')
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(112): com_wiris_util_sys_Store->write('mml=<math xmlns...')
#4 /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...')
#5 /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)
#6 /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)
#7 {main}</pre>
            equals fraction numerator sin squared straight B plus sinB space sin left parenthesis straight C minus straight A right parenthesis over denominator 2 sinB space sinC end fraction equals fraction numerator sinB space plus space sin left parenthesis straight C minus straight A right parenthesis over denominator 2 sinC end fraction
           equals fraction numerator sin left parenthesis straight pi minus straight C plus straight A right parenthesis plus sin left parenthesis straight C minus straight A right parenthesis over denominator 2 sinC end fraction equals fraction numerator sin left parenthesis straight C plus straight A right parenthesis plus sin left parenthesis straight C minus straight A right parenthesis over denominator 2 sinC end fraction
          space space equals fraction numerator 2 sinC space cosA over denominator 2 sinC end fraction equals cosA        [∵ sin (A + B) + sin (A - B) = 2 sinA cosB]
            = L.H.S.
           

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