Consider the following relations among the angles `alpha,beta and gamma` made by a vector with the coordinate axes
I. `cos^2alpha+cos2beta+cos2gamma=-1`
II. `sin^2alpha+sin^2beta+sin^2gamma=1`
Which of the above is/are correct?

Options

A. only I
B. Only II
C. Both I and II
D. Neither I or II

Answer

A

Explanation

We have,
`cos^2alpha+cos^2beta+cos^2gamma=1 …(i)`
`rArr 2cos^2alpha^2beta+2cos^2gamma=2`
`rArr 2cos^2alpha-1+2cos^2beta-1+2cos^2gamma-1=2-3`
`rArr cos 2alpha +cos 2 beta+cos 2 gamma=-1`
Hence statement -I is correct.
and now from (i),
`1-sin^2alpha+1-sin^2beta+1-sin^2gamma=1`
`rArr sin^2alpha+sin^2beta+sin^2gamma =2`
Hence, only statement I is correct.