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  4. The value of cos 2( (π /4)+θ )- sin 2( (π /4)-θ ) is

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Q. The value of $ {{\cos }^{2}}\left( \frac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \frac{\pi }{4}-\theta \right) $ is

J & K CETJ & K CET 2007Trigonometric Functions

Solution:

We know that, $ {{\cos }^{2}}\,(A)-{{\sin }^{2}}(B) $
$ =\cos (A+B)\,\cos \,(A-B) $
$ \therefore $ $ {{\cos }^{2}}\,\left( \frac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \frac{\pi }{4}-\theta \right) $
$ =\cos \,\left( \frac{\pi }{4}+\theta +\frac{\pi }{4}-\theta \right)\,\,\cos \,\left( \frac{\pi }{4}+\theta -\frac{\pi }{4}+\theta \right) $
$ =\cos \,\left( \frac{\pi }{2} \right)\,\cos \,(2\,\theta )=0 $