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

Question

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 $80$.

Answer 1

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 $x = 1$.

Answer 2

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 $\sin^22x=1\iff\cos2x=0\iff a=b$