Principle of Mathematical Induction

Principle of Mathematical Induction

Question

In any ΔABC comma if straight a space cos squared straight C over 2 plus straight c space cos squared straight A over 2 space equals space fraction numerator 3 straight b over denominator 2 end fraction comma show that the sides of the triangles are in A.P.

Answer

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#6 {main}</pre>
acos squared straight C over 2 plus ccos squared straight A over 2 space equals space fraction numerator 3 straight b over denominator 2 end fraction
   (Given)                                             ...(i)
Now,              1 plus cosC space equals space 2 cos squared straight C over 2
rightwards double arrow                     space space space cos squared straight C over 2 space equals space fraction numerator 1 plus cosC over denominator 2 end fraction
Similarly, 
                                 cos squared straight A over 2 space equals space fraction numerator 1 plus cosA over denominator 2 end fraction
Putting in (i), we get
straight a open parentheses fraction numerator 1 plus cosC over denominator 2 end fraction close parentheses plus straight c open parentheses fraction numerator 1 plus cosA over denominator 2 end fraction close parentheses space equals space fraction numerator 3 straight b over denominator 2 end fraction
rightwards double arrow   space space space straight a plus straight a space cosC space plus space straight c space plus space straight c space cosA space equals space 3 straight b
rightwards double arrow   straight a plus straight c plus left parenthesis straight a space cos space straight C space plus space straight c space cosA right parenthesis space equals space 3 straight b
By projection formula,
straight a space cosC space plus space straight c space cosA space equals space straight b
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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)
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#6 {main}</pre>
rightwards double arrow                                 space space space space space space 2 straight b space equals space straight a plus straight c space
rightwards double arrow                                      <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)
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#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...')
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#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>
Hence, the sides a, b, c of trianlge ABC are in A.P.

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