A) 1 |
B) 2 |
C) 3 |
D) 4 |
B) 2 |
\( \Large log\ 0.5 \sin x + log\ 0.5 \cos x = 1 \)
=> \( \Large log\ 0.5 \sin x \cos x = 1 \)
=> \( \Large \sin x \cos x = \frac{1}{2} => \sin 2x = 1 \)
=> \( \Large 2x = n \pi + \left(-1\right)^{n}\frac{ \pi }{2} => x = \frac{n \pi }{2}+ \left(-1\right)^{n}\frac{ \pi }{4} \)
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.