Question

$x=k$ is the only solution satisfying the equation $1+{\sin }^{2}ax=\cos x$ , where $a$ is irrational. Find the value of $k$.

Answer 1

The given equation is $1+{\sin }^{2}ax=\cos x$ ...(1)
We have $1+{\sin }^{2}x\ge 1$ and $\cos x\le 1$
Therefore equation (1) has a solution only if $\cos x=1$
and $1+{\sin }^{2}ax=1\Rightarrow {\sin }^{2}ax=0\Rightarrow \sin ax=0$
Now $\cos x=1$ gives $x=2n\pi ,n\in I$
and $\sin ax=0$ gives $x=m\pi$ $\Rightarrow x=\frac{m\pi }{a},m\in I$
The equation (1) has a solution only if for same value of $m$ and $n$, we have
$\frac{m\pi }{a}=2n\pi$ or $2na=m$ ...(2)
But $a$ is irrational and $m$ and $n$ are integer
Therefore (2) is possible only if $m=0=n$
But then we have $x=0$
Hence $x=0$ is the only solution satisfying the given equation