A line makes an angel `theta` with each of the x-and z-axes. If the angel `beta,` which it makes with the y-axis, is such that `sin^2beta=3sin^2theta,t h e ncos^2theta` equals a. `2/3` b. `1/5` c. `3/5` d. `2/5`

Options

A. `2/3`
B. `1/(sqrt(2))`
C. `1/2`
D. Data inappropriate

Answer

C

Explanation

A line makes angle `theta` with x-axis and z-axis and `beta` with y-axis.
`:.l=costheta,m=cosbeta,n=costheta`
`:'l^(2)+m^(2)+n^(2)=1`
`:. cos^(2)theta+cos^(2)beta+cos^(2)theta=1`
`implies2cos^(2)theta=1-cos^(2)beta`
`implies2cos^(2)theta=sin^(2)beta`
But, it is given
`sin^(2)beta=2sin^(2)theta`
Form Eqs i and ii we get
`2sin^(2)theta=2cos^(2)theta`
`impliestan^(2)theta=1impliescos^(2)theta=1/2`