This is a beautiful problem from ISI MStat 2019 PSA problem 12 based on finding the domain of the function. We provide sequential hints so that you can try.
What is the set of numbers \(x\) in \( (0,2 \pi)\) such that \(\log \log (\sin x+\cos x)\) is well-defined?
Domain
Basic inequality
Trigonometry
Answer: is \( (0,\frac{\pi}{2}) \)
ISI MStat 2019 PSA Problem 12
Pre-college Mathematics
\(logx\) is defined for \( x \in (0,\infty)\).
\(sinx+cosx > 0\).
\(log(sinx+cosx) > 0 \Rightarrow sinx + cosx > 1\)
\( sin(x+\frac{\pi}{4}) > \frac{1}{\sqrt{2}}\)
For \(y\) in \( (0,2 \pi)\) , \(siny > \frac{1}{\sqrt{2}} \iff \frac{\pi}{4} < y < \frac{3\pi}{4 } \)
Hence we have \( 0< x < \frac{\pi}{2 } \) .
