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I tried solving the question by taking common terms and simplifying. But I am stuck. How to find L ?

By the way Answer is 8

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    Setting $x = \cos^2\alpha$ and $y = \cos^2\beta$, it seems that [WolframAlpha](http://www.wolframalpha.com/input/?i=plot+(x%5E3+-+x%5E3y+-+xy%5E3+%2B+y%5E3)%2F(x%5E2y%5E2+-+x%5E3y%5E2+-+x%5E2y%5E3+%2B+x%5E3y%5E3)+for+x+from+0+to+1,+y+from+0+to+1) disagrees with the value $8$. Although $8$ is the _minimal_ value. – 2017-01-25
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    Isn't it given some other relation between $\;\alpha,\,\beta\;$ ? – 2017-01-25
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    I have Uploaded the full question. Please see – 2017-01-26
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    @GeorgesElencwajg hey, please take a look again I have uploaded the full question. – 2017-01-27

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You need to use $\sin^2 x + \cos^2 x=1$ many times. For example the first and third terms in the numerator give $$\cos^6 \beta - \cos^6 \beta \cos^2 \alpha=\cos^6 \beta \sin^2 \alpha$$ Then do the second and fourth terms the same way. You also need the sum rule $\sin (x+y)=\sin x \cos y + \cos x \sin y.$ I don't know if that gets you all the way there, but it goes a long way.

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    It is not taking me anywhere close to the answer. – 2017-01-26
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    The second and fourth terms in the numerator do this, as do two pairs in the denominator. You should do this and show where you get, then where you are stuck. It is clearly progress as you go from eight to four terms total. – 2017-01-26