The equation `1+sin^2 ax = cos x` has a unique solution then a is

Options

A. equal to 1
B. a rational number
C. an irrational number
D. none of these

Answer

B

Explanation

We have,
`1 + "sin"^(2) ax = "cos"x`
We observe that
` 1+ "sin"^(2) ax ge 1 " and cos" x le 1 "for all "x.`
Therefore, ` 1 +"sin"^(2) ax = "cos" x` is valid if each side is equal to 1.
i.e `1 + "sin"^(2) ax = 1 " and cos" x = 1`
`rArr "sin"^(2) ax = 0 " and cos" x = 1`
Now, `"cos" x = 1 rArr x = 2n pi, n in Z`.
These values of x will satisfy `"sin"^(2) ax =0 "if" a in Q`.
Thus, the given equation will have infinitely many solutions if `a in Q`.