Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove that in triangle ABC,

straight a space left parenthesis cosC space minus space cosB right parenthesis space equals space 2 left parenthesis straight b minus straight c right parenthesis space cos squared open parentheses straight A over 2 close parentheses

Answer

L.H.S. = a (Cos C - Cos B)
          = straight a open square brackets fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction minus fraction numerator straight c squared plus straight a squared minus straight b squared over denominator 2 ac end fraction close square brackets
          = 1 half open square brackets fraction numerator straight a squared plus straight b squared minus straight c squared over denominator straight b end fraction minus fraction numerator straight c squared plus straight a squared minus straight b squared over denominator straight c end fraction close square brackets
          = 1 half open square brackets fraction numerator straight a squared straight c plus straight b squared straight c minus straight c cubed minus bc squared minus ba squared plus straight b cubed over denominator bc end fraction close square brackets
          = 1 half open square brackets fraction numerator left parenthesis straight b cubed minus straight c cubed right parenthesis plus left parenthesis straight b squared straight c minus bc squared right parenthesis plus left parenthesis straight a squared straight c minus straight a squared straight b right parenthesis over denominator bc end fraction close square brackets
           = space space 1 half left parenthesis straight b minus straight c right parenthesis space open square brackets fraction numerator straight b squared plus straight c squared plus bc plus bc minus straight a squared over denominator bc end fraction close square brackets space equals space 1 half left parenthesis straight b minus straight c right parenthesis open square brackets fraction numerator left parenthesis straight b plus straight c right parenthesis squared minus straight a squared over denominator bc end fraction close square brackets
            = <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>                                                 ...(i)
Now,          straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction
rightwards double arrow           2s = a + b + c
rightwards double arrow         2s - 2a = a + b + c - 2a = b + c - a
By using these values in (i), we get
                 L.H.S. = 1 half fraction numerator left parenthesis straight b minus straight c right parenthesis space left parenthesis 2 straight s minus 2 straight a right parenthesis space left parenthesis 2 straight s right parenthesis over denominator bc end fraction space equals space 2 space left parenthesis straight b minus straight c right parenthesis. space fraction numerator straight s left parenthesis straight s minus straight a right parenthesis over denominator bc end fraction
                       space space space space equals space 2 space left parenthesis straight b space minus straight c right parenthesis. space open parentheses square root of fraction numerator straight s left parenthesis straight s minus straight a right parenthesis over denominator bc end fraction end root close parentheses squared
But,      square root of fraction numerator straight s left parenthesis straight s minus straight a right parenthesis over denominator bc end fraction end root space equals space cos straight A over 2
∴             L.H.S. = 2 left parenthesis straight b minus straight c right parenthesis space cos squared straight A over 2 equals space straight R. straight H. straight S.
Hence,  <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>

More Chapters from Principle of Mathematical Induction