\( \Large log\ 0.5 \sin x + log\ 0.5 \cos x = 1 \)

=> \( \Large log\ 0.5 \sin x \cos x = 1 \)

=> \( \Large \sin x \cos x = \frac{1}{2} => \sin 2x = 1 \)

=> \( \Large 2x = n \pi + \left(-1\right)^{n}\frac{ \pi }{2} => x = \frac{n \pi }{2}+ \left(-1\right)^{n}\frac{ \pi }{4} \)

Since \( \Large log_{0.5} \sin x\ and\ log_{0.5}.\cos x\ are\ real \)

\( \Large \therefore \  \sin x \  and \  \cos x \  must \ lie\ in\ first\ quardrant \)

\( \Large \therefore x = \frac{ \pi }{4}, -2 \pi + \frac{ \pi }{4} \)

Hence, the number of solutions is 2.