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 $80$.
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 $x = 1$.
Let $a=\sec^2x,b=\csc^2x\implies a+b=ab$
The equality occurs if $\sin^22x=1\iff\cos2x=0\iff a=b$