In ` A B C` Prove that `cos^2A/2+cos^2B/2+cos^2C/2lt=9/4dot` If `cos^2A/2+cos^2B/2+cos^2C/2=y(x^2+1/(x^2))` then find the maximum value of `ydot`

Answer

`9//8`

Explanation

(a) In `triangle ABC`, we know that
`cosA+cosB+cos Cle(3)/(2)`
`cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)`
`=(1+cosA)/(2)+(1+cosB)/(2)+(1+cosC)/(2)`
`=(3)/(2)+(cos A+cosB+cosC)/(2)le(3)/(2)+(3)/(4)` (using eq. i)
`therefore cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)le(9)/(4)`.
(b) `cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)=y(x^(2)+(1)/(x^(2)))`
`therefore y(x^(2)+(1)/(x^(2)))le(9)/(4)`.
`therefore yle(9)/(4(x^(2)+(1)/(x^(2)))`
Now `x^(2)+(1)/(x^(2))ge2`
`therefore yle(9)/(8)`.
thus, maximum value of `y` is `(9)/(8)`.