Principle of Mathematical Induction

Question

Prove that $sin space 2 straight x space equals space 2 space sinx space cosx space equals space fraction numerator 2 space tanx over denominator 1 plus tan squared straight x end fraction.$

Solution

We know that
 $sin left parenthesis straight alpha plus straight beta right parenthesis space equals space sinα space cosβ space plus space cosα space sinβ$ ...(i)
Replacing $space space space straight alpha space equals space straight beta space equals space straight x$ in (i), we get
∴ sin (2x) = sinx. cosx + cosx. sinx
∴ sin 2x = 2 sinx cosx ...(A)
Also, $sin 2 straight x space equals space fraction numerator 2 sinx space cosx over denominator 1 end fraction equals fraction numerator 2 sinx space cosx over denominator cos squared straight x plus sin squared straight x end fraction$
Dividing numerator and denominator by $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ we get
 $sin 2 straight x space equals space fraction numerator begin display style fraction numerator 2 sinx space cosx over denominator cos squared straight x end fraction end style over denominator begin display style fraction numerator cos squared straight x plus sin squared straight x over denominator cos squared straight x end fraction end style end fraction space equals space fraction numerator 2 tanx over denominator begin display style fraction numerator cos squared straight x over denominator cos squared straight x end fraction end style plus begin display style fraction numerator sin squared straight x over denominator cos squared straight x end fraction end style end fraction$
$rightwards double arrow$ $sin space 2 straight x space equals space fraction numerator 2 space tanx over denominator 1 plus tan squared straight x end fraction$ ...(B)
From (A) and (B), we have

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

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Principle Of Mathematical Induction

Prove that $sin space 2 straight x space equals space 2 space sinx space cosx space equals space fraction numerator 2 space tanx over denominator 1 plus tan squared straight x end fraction.$

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