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જો $n$ એ પૂર્ણાક હોય તો સમીકરણ $\cos x - \sin x = \frac{1}{{\sqrt 2 }}$ નો વ્યાપક ઉકેલ મેળવો.
$x = 2n\pi - \frac{\pi }{{12}}$ અથવા $x = 2n\pi + \frac{{7\pi }}{{12}}$
$x = n\pi \pm \frac{\pi }{{12}}$
$x = 2n\pi + \frac{\pi }{{12}}$ અથવા $x = 2n\pi - \frac{{7\pi }}{{12}}$
$x = n\pi + \frac{\pi }{{12}}$ અથવા $x = n\pi - \frac{{7\pi }}{{12}}$
Solution
(c) Given equation is,$\cos x – \sin x = \frac{1}{{\sqrt 2 }}$
Dividing equation by $\sqrt 2 $,
$\frac{1}{{\sqrt 2 }}\cos x – \frac{1}{{\sqrt 2 }}\sin x = \frac{1}{2}$
$\cos \left( {\frac{\pi }{4} + x} \right) = \cos \frac{\pi }{3}$.
Hence, $\frac{\pi }{4} + x = 2n\pi \pm \frac{\pi }{3}$
$x = 2n\pi + \frac{\pi }{3} – \frac{\pi }{4} = 2n\pi + \frac{\pi }{{12}}$
or $x = 2n\pi – \frac{\pi }{3} – \frac{\pi }{4} = 2n\pi – \frac{{7\pi }}{{12}}$.
