Principle of Mathematical Induction

Question

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$space space tan left parenthesis straight x minus straight y right parenthesis space plus space tan left parenthesis straight y minus straight z right parenthesis plus tan left parenthesis straight z minus straight x right parenthesis space equals space tan left parenthesis straight x minus straight y right parenthesis. space tan left parenthesis straight y minus straight z right parenthesis. space tan left parenthesis straight z minus straight x right parenthesis$

Solution

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

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