Trigonometric identies equation help

Not exactly sure how to get rid of the power of 8 using the identies, splitting it up individually into cos doesn't seem like it'll work

Not exactly sure how to get rid of the power of 8 using the identies, splitting it up individually into cos doesn't seem like it'll work

\sin^8 = (1-\cos^2 x)^4

then let

u = \cos x

and get a polynomial in

u

.

What'd I do is by inspection

x = \pi

and then prove that

(3 +\cos x)^2

is always bigger than

4 - 2\sin^8 x

other than the touching at

x = \pi

Looking at it again, this is very easy.

2 \leq 3 + \cos x \leq 4

whilst

2 \leq 4 - 2\sin^8 x \leq 4

Ahh okay so the one solution being pi
Then just proving that the cos graph is always bigger and not touching the sin graph again

Ahh okay so the one solution being pi
Then just proving that the cos graph is always bigger and not touching the sin graph again

Yep!

The main issue as I cannot use a calculator is factorising this or using another method to prove that they only meet at pi

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