Consider the following statements in respect of the determinant `|[cos^2(alpha/2),sin^2(alpha/2)],[sin^2(beta/2),cos^2(beta/2)]|` where `alpha,beta` are complementary angles

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

C

Explanation

`alpha+beta=90^(@)`
`|(cos^(2).(alpha)/(2),sin^(2).(alpha)/(2)),(sin^(2).(beta)/(2),cos^(2).(beta)/(2))|`
`=cos^(2).(alpha)/(2)cos.(beta)/(2)-sin^(2).(alpha)/(2)sin^(2).(beta)/(2)`
`=(cos.(alpha)/(2)cos.(beta)/(2)+sin.(alpha)/(2)sin.(beta)/(2))(cos.(alpha)/(2)cos.(beta)/(2)-sin.(alpha)/(2)sin.(beta)/(2))`
`=cos.((alpha-beta))/(2)xxcos((alpha+beta))/(2)`
`=cos((alpha-beta))/(2)xxcos((90^(@)))/(2)`
`cos((alpha-beta))/(2)xx(1)/(sqrt2)`
Maximum value of `cos((alpha-beta)/(2))` is 1. So maximum value of determinent is `((1)/(sqrt2))`
So both 1 and 2 are correct.