# Find the least value of $\sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$

Find the least value of $$\sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$$ I tried AM greater than equal to GM But that's for finding maximum value. This can probably be solved with calculus but I don't know for some reason I can't find the answer . Which is $$80$$.

• Add 1 and subtract 1 distribute the factorizable terms trust me it will help – Aditya Garg Mar 29 at 8:23

Hint: use trig identities to write it in terms of $$\tan\theta$$. Then use the fact that the minimum value of $$x + 1/x$$ occurs at $$x = 1$$.

• This hint makes the problem very simple, for sure ! $\to +1$ – Claude Leibovici Mar 29 at 8:21

Let $$a=\sec^2x,b=\csc^2x\implies a+b=ab$$

$$a^3+b^3+a^3b^3=(a+b)^3-3ab(a+b)+a^3b^3=a^2b^2(2ab-3)$$

Now $$ab=\dfrac4{\sin^22x}\ge4$$

The equality occurs if $$\sin^22x=1\iff\cos2x=0\iff a=b$$

• funny solution :-) thank you! – Math-fun Mar 29 at 8:42
• – lab bhattacharjee Mar 29 at 8:54
• thank you! looked interesting – Math-fun Mar 29 at 9:02