Principle of Mathematical Induction

Principle of Mathematical Induction

Question

If tan space straight alpha space equals space fraction numerator straight m over denominator straight m plus 1 end fraction and  <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)
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#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)
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#6 {main}</pre> where space space straight m not equal to space minus 1 comma space space fraction numerator negative 1 over denominator 2 end fraction prove that

straight alpha plus straight beta space equals space straight pi over 4.

Answer

Here, tanα space equals space fraction numerator straight m over denominator straight m plus 1 end fraction comma space space tanβ space equals space fraction numerator 1 over denominator 2 straight m plus 1 end fraction
    space space space space space space tan open parentheses straight alpha plus straight beta close parentheses space equals space fraction numerator tanα plus tanβ over denominator 1 minus tanα space tanβ end fraction space equals space fraction numerator begin display style fraction numerator straight m over denominator straight m plus 1 end fraction plus fraction numerator 1 over denominator 2 straight m plus 1 end fraction end style over denominator 1 minus begin display style fraction numerator straight m over denominator straight m plus 1 end fraction end style. begin display style fraction numerator 1 over denominator 2 straight m plus 1 end fraction end style end fraction
                       equals fraction numerator begin display style fraction numerator 2 straight m squared plus straight m plus straight m plus 1 over denominator left parenthesis straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 1 right parenthesis end fraction end style over denominator begin display style fraction numerator 2 straight m squared plus 2 straight m plus straight m plus 1 minus straight m over denominator left parenthesis straight m plus 1 right parenthesis left parenthesis 2 straight m plus 1 right parenthesis end fraction end style end fraction space equals space fraction numerator 2 straight m squared plus 2 straight m plus 1 over denominator 2 straight m squared plus 2 straight m plus 1 end fraction equals 1
rightwards double arrow                tan left parenthesis straight alpha plus straight beta right parenthesis space equals space 1 space equals space tan straight pi over 4
rightwards double arrow                       straight alpha plus straight beta space equals space straight pi over 4

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