Question

The equation $1+{\sin }^2$ ax $=$ cosx has a unique solution then a is _____________.

• A. rational
• B. irrational
• C. integer
• D. whole number

Answer 1

The correct option is A rational

Given $1+{\sin }^{2}ax=\cos x$

$⟹1-\cos x=-{\sin }^{2}ax$

$⟹2{\sin }^2\frac{x}{2}=-{\sin }^{2}ax$ $(\because 1-\cos 2A=2{\sin }^{2}A)$

We know that

$2{\sin }^2\frac{x}{2}\ge 0$ and $-{\sin }^{2}ax\le 0$

For the above equation to have a unique solution

So ${\sin }^{2}ax=0⟹ax=n\pi ⟹x=\frac{n\pi }{a}$ where $n$ is an integer

and also ${\sin }^2\frac{x}{2}=0⟹x=2k\pi$ where $k$ is an integer

$⟹\frac{n\pi }{a}=2k\pi ⟹a=\frac{n}{2k}$ where $n,k$ are integers

So $a$ is a rational number