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  4. If 2 cos α =a+( (1/a) ) and 2 cos β =b+( (1/b) ), then ab+( (1/ab) ) is equal to

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Q. If $ 2\cos \alpha =a+\left( \frac{1}{a} \right) $ and $ 2\cos \beta =b+\left( \frac{1}{b} \right), $ then $ ab+\left( \frac{1}{ab} \right) $ is equal to

Rajasthan PETRajasthan PET 2004

Solution:

Given, $ 2\cos \alpha =a+\frac{1}{a} $
$ \Rightarrow $ $ {{a}^{2}}-2a\cos \alpha +1=0 $ ...(i)
and $ 2\cos \beta =b+\frac{1}{b} $
$ \Rightarrow $ $ {{b}^{2}}-2b\cos \beta +1=0 $ ...(ii)
From Eq. (i),
$ a=\frac{2\cos \alpha \pm \sqrt{4{{\cos }^{2}}\alpha -4}}{2} $
$ \Rightarrow $ $ a=\frac{2\cos \alpha \pm 2i\sin \alpha }{2} $
$ \Rightarrow $ $ a=\cos \alpha \pm i\sin \alpha $
Similarly, $ b=cos\text{ }\beta \pm i\text{ }sin\text{ }\beta $
$ \therefore $ $ ab=(\cos \alpha \pm i\sin \alpha )(\cos \beta \pm i\sin B) $
$ =\cos (\alpha +\beta )\pm i\sin (\alpha +\beta ) $
and $ \frac{1}{ab}=\cos (\alpha +\beta )\mp i\sin (\alpha +\beta ) $
Hence, $ ab+\frac{1}{ab}=2\cos (\alpha +\beta ) $