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

logosomatw

logosomatw

Answered question

2022-01-25

Find the least value of sec6x+csc6x+sec6xcsc6x

Answer & Explanation

SaphypycleDapc5

SaphypycleDapc5

Beginner2022-01-26Added 11 answers

Let a=sec2x,b=csc2xa+b=ab
a3+b3+a3b3=(a+b)33ab(a+b)+a3b3=a2b2(2ab3)
Now ab=4sin22x4
The equality occurs if sin22x=1cos2x=0a=b
Kudusind

Kudusind

Beginner2022-01-27Added 11 answers

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

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