If for $ x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1 $ and $ \log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), n0 $ then the value of $ n $ is equal to: [JEE Main 2021, 16 Mar (Shift 1)]

Question

If for $ x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1 $ and $ \log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), n>0 $ then the value of $ n $ is equal to: [JEE Main 2021, 16 Mar (Shift 1)]

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If for \( x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1 \) and \( \log _{10}(\sin x+\cos x)…

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