Principle of Mathematical Induction

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Question

Solve the equation:

$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

Solution

Given equation is:
$tan left parenthesis straight x plus straight a right parenthesis space. space tan left parenthesis straight x plus straight b right parenthesis space plus space tan left parenthesis straight x plus straight b right parenthesis. space tan left parenthesis straight x plus straight c right parenthesis space plus space tan left parenthesis straight x plus straight c right parenthesis space tan space left parenthesis straight x plus straight a right parenthesis space equals space 1$
$tan space left parenthesis straight x plus straight a right parenthesis space tan space left parenthesis straight x plus straight b right parenthesis space plus tan space left parenthesis straight x plus straight b right parenthesis space tan space left parenthesis straight x plus straight c right parenthesis space equals space 1 minus tan space left parenthesis straight x plus straight c right parenthesis space tan space left parenthesis straight x plus straight a right parenthesis$
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$ $tan space left parenthesis straight x plus straight b right parenthesis space open curly brackets fraction numerator tan space left parenthesis straight x plus straight a right parenthesis space plus space tan space left parenthesis straight x plus straight c right parenthesis over denominator 1 minus tan space left parenthesis straight x plus straight a right parenthesis space tan space left parenthesis straight x plus straight c right parenthesis end fraction close curly brackets space equals space 1$
$rightwards double arrow$ $tan space left parenthesis straight x plus straight b right parenthesis space tan space open curly brackets open parentheses straight x plus straight a close parentheses plus left parenthesis straight x plus straight c right parenthesis close curly brackets space equals space 1$
$rightwards double arrow$ $tan space left parenthesis straight x plus straight a plus straight x plus straight c right parenthesis space equals space fraction numerator 1 over denominator tan space left parenthesis straight x plus straight b right parenthesis end fraction$ $rightwards double arrow$ $tan space left parenthesis 2 straight x plus straight a plus straight c right parenthesis space equals space cot space left parenthesis straight x plus straight b right parenthesis$
$rightwards double arrow$ $tan space left parenthesis 2 straight x plus straight a plus straight c right parenthesis space equals space tan open parentheses straight pi over 2 minus left parenthesis straight x plus straight b right parenthesis close parentheses$ $rightwards double arrow$ $2 straight x plus straight a plus straight c space equals space nπ space plus space straight pi over 2 minus left parenthesis straight x plus straight b right parenthesis comma space straight n space element of space straight Z$
$rightwards double arrow$ $2 straight x plus straight a plus straight c space equals space nπ space plus space straight pi over 2 minus straight x minus straight b comma space straight n element of straight Z$ $rightwards double arrow$ $space space space 3 straight x space equals space nπ plus straight pi over 2 minus straight b minus straight a minus straight c$
$rightwards double arrow$ $3 straight x space equals space nπ space plus space straight pi over 2 minus left parenthesis straight a plus straight b plus straight c right parenthesis comma space straight n space element of space straight Z$ $rightwards double arrow$ $straight x space equals space nπ over 3 plus straight pi over 6 minus open parentheses fraction numerator straight a plus straight b plus straight c over denominator 3 end fraction close parentheses comma space straight n space element of space straight Z$

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Principle Of Mathematical Induction

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If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

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