`x=sqrt(a^2cos^2alpha+b^2sin^2alpha)+sqrt(a^2sin^2alpha+b^2cos^2alpha)` then `x^2=a^2+b^2+2sqrt(p(a^2+b^2)-p^2),` where p can be is equal to

Options

A. `alpha^(2)cos^(2)alpha+b^(2)sin^(2)alpha`
B. `alpha^(2)sin^(2)alpha+b^(2) cos^(2)alpha`
C. `(1)/(2) [alpha^(2)+b^(2)+(alpha^(2)-b^(2))cos2alpha[`
D. `(1)/(2)[a^(2)+b^(2)-(a^(2)-b^(2))cos2alpha]`

Answer

A, B, C, D

Explanation

`" "x = sqrt (a^(2)cos^(2) alpha + b^(2) sin^(2) alpha) + sqrt(a^(2) sin^(2) alpha + b^(2) cos^(2) alpha)`
`rArr x^(2) = a^(2) + b^(2) + 2 sqrt((a^(2) cos ^(2) alpha + b^(2) sin^(2) alpha) ( a^(2)sin^(2) alpha + b^(2) cos^(2)alpha))`
`= a^(2) + b^(2) 2k`,
where `k = sqrt([(a^(2) + b^(2)) - (a^(2) sin^(2) alpha + b^(2) cos^(2) alpha )] xx (a^(2) sin^(2)alpha + b^(2) cos^(2) alpha))`
`therefore x = a^(2) + b^(2) + 2 sqrt((a^(2) + b^(2)) p-p^(2))`
where `p = a^(2) sin ^(2) alpha + b^(2) cos^(2) alpha`
`" " = (alpha^(2))/(2) (1-cos 2alpha) + (b^(2)) /(2) (1+ cos2alpha)`
`" " = (1)/(2)[a^(2)+ b^(2) - (a^(2)-b^(2)) cos 2alpha]`
Also, `x^(2) = a^(2) + b^(2) + 2sqrt((a^(2) cos^(2) alpha+ b^(2) sin^(2) alpha) (a^(2) sin^(2) alpha + b^(2) cos^(2)alpha))`
`" " = a^(2) + b^(2) +2 sqrt((a^(2) + b^(2)) p-p^(2))`
where , `p =a^(2) cos^(2) alpha + b^(2) sin^(2) alpha`
or `p = (1)/(2) [a^(2) + b^(2) + (a^(2) - b^(2)) cos 2alpha]`