Principle Of Mathematical Induction

Question
CBSEENMA11014298

Prove the following:

cos left parenthesis 2 straight pi minus straight x right parenthesis space equals space cosx

sin left parenthesis 2 straight pi minus straight x right parenthesis equals negative sinx

tan left parenthesis 2 straight pi minus straight x right parenthesis space equals space minus tanx











Solution

We know that:
cos(x - y) = cosx cosy + sinx siny                                 ...(i)
sin (x - y) = sinx cosy - cosx siny                                  ...(ii)
   cos left parenthesis straight pi right parenthesis space equals space 1                                                              ...(iii)
    <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>                                                              ...(iv)
Replacing x by 2 straight pi and y by x in (i), we get
          cos left parenthesis 2 straight pi minus straight x right parenthesis space equals space cos space 2 straight pi space cosx space plus space sin space 2 straight pi space sinx
                              equals 1 space left parenthesis cosx right parenthesis space plus space 0 space left parenthesis sinx right parenthesis              [By using (iii) and (iv)]
                               = cos x
Hence,     cos left parenthesis 2 straight pi minus straight x right parenthesis space equals space cosx
Replacing x by 2 straight pi and y by x in (iii), we get
rightwards double arrow               sin space left parenthesis 2 straight pi minus straight x right parenthesis space equals space sin space 2 straight pi space cosx space minus space cos space 2 straight pi space sinx space equals space 0 space left parenthesis cosx right parenthesis space minus space left parenthesis 1 right parenthesis space sinx
                                       = - sinx                           [By using (iii) and (iv)]
Hence,    sin left parenthesis 2 straight pi minus straight x right parenthesis space equals space minus sinx
              tan left parenthesis 2 straight pi minus straight x right parenthesis space equals space fraction numerator sin left parenthesis 2 straight pi minus straight x right parenthesis over denominator cos left parenthesis 2 straight pi minus straight x right parenthesis end fraction equals fraction numerator negative sinx over denominator cosx end fraction equals negative tanx
Hence,     tan left parenthesis 2 straight pi minus straight x right parenthesis space equals space minus tanx
         


Sponsor Area