The number of solution the equation `cos(theta).cos(pitheta)=1` has 0 (b) 1 (c) 4 (d) 2

Options

A. 0
B. 2
C. 1
D. infinite

Answer

C

Explanation

`cos (theta) cos (pi theta)=1`
`rArr cos (theta)=1 and cos (pi theta)=1` ...(i)
or `cos (theta) =-1 and cos (pi theta)=-1` ...(ii)
If `cos (theta)=1 rArr theta=2 mpi`
and `cos (pi theta) =1 rArr theta =2k, k in Z`
which is possible only when `theta=0`.
Equation (ii) is not possible for any `theta` as for `cos (theta)=-1, theta` should be odd multiple of `pi`, i.e., irrational and for `cos(pi theta)=-1, theta` should be odd integer, i.e., rational.
Both the conditions cannot be satisfied.
Therefore, `theta=0` is the only solution.