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Principle Of Mathematical Induction

Question
CBSEENMA11014326

Find general value of x, when cos x = cos y

Solution

We have cosx = cosy               or    cosx - cosy = 0
rightwards double arrow negative 2 sin open parentheses fraction numerator straight x plus straight y over denominator 2 end fraction close parentheses sin open parentheses fraction numerator straight x minus straight y over denominator 2 end fraction close parentheses equals 0
rightwards double arrow   either       space sin fraction numerator straight x plus straight y over denominator 2 end fraction equals 0          or     sin fraction numerator straight x minus straight y over denominator 2 end fraction equals 0
rightwards double arrow   either          fraction numerator straight x plus straight y over denominator 2 end fraction space equals space kπ        or      fraction numerator straight x minus straight y over denominator 2 end fraction space equals space kπ comma where straight k space element of space straight Z
rightwards double arrow   either         space space straight x plus straight y equals 2 kπ         or    straight x minus straight y equals 2 kπ comma   where straight k space element of space straight Z
rightwards double arrow   either              <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)
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#6 {main}</pre>     or   straight x minus straight y equals 2 kπ plus straight y, where straight k element of space straight Z
rightwards double arrow                         straight x equals 2 kπ plus-or-minus straight y comma where straight k space element of space straight Z
Hence, the general solution is straight x equals 2 nπ plus-or-minus straight y comma straight n element of space straight Z
COROLLARY: If secx = secy rightwards double arrow cosx = cosy
So, when secx = secy, the general solution of straight x equals 2 nπ plus-or-minus straight y comma where straight n element of space straight Z