Mathematics - Trigonometric Equation Question with Solution | TestHub

The equation $1+{\sin }^2(ax)=\cos x$ implies $1+{\sin }^2(ax)\ge 1$ and $\cos x\le 1$. For equality, both sides must be 1. Thus, $\sin (ax)=0$ and $\cos x=1$. This means $x=2n\pi$ and $ax=k\pi$ for integers $n,k$. Substituting $x$ gives $2an=k$. Since $a$ is irrational, $n$ must be 0, implying $x=0$. Hence, there is only one solution.