If for x ∈ (0, π/2),

$x \in\left(0, \frac{\pi}{2}\right)$

$\log _{10} \sin x+\log _{10} \cos x=-1$

$\Rightarrow \quad \log _{10} \sin x \cdot \cos x=-1$

$\Rightarrow \quad \sin x \cdot \cos x=\frac{1}{10}$........(1)

$\log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right)$

$\Rightarrow \quad \sin x+\cos x=10^{\left(\log _{10} \sqrt{n}-\frac{1}{2}\right)}=\sqrt{\frac{n}{10}}$

by squaring

$1+2 \sin x \cdot \cos x=\frac{n}{10}$

$\Rightarrow 1+\frac{1}{5}=\frac{\mathrm{n}}{10} \quad \Rightarrow \mathrm{n}=12$