Question

If $lo{g}_{0.5}cosx=1-lo{g}_{0.5}$ sin x, then the number of solution in $[-2\pi ,2\pi ]$ is

Answer 1

$lo{g}_{0.5}cosx=1-lo{g}_{0.5}sinx$
$lo{g}_{0.5}cosxsinx=1$
$=lo{g}_{0.5}0.5$
$[\because lo{g}_{a}a=1]$

$\Rightarrow sinxcosx=0.5$
$2sinxcosx=1$
$sin2x=1$
$\Rightarrow 2x=\frac{\pi }{2}+2n\pi$
$x=\frac{\pi }{4}+n\pi$
log is defined only for positive.
$\therefore x=\frac{\pi }{4},\frac{-7\pi }{4}isasolution$