Question

The solution of ${\log }_{0.5}\cos x=1-{\log }_{0.5}\sin x$, where $x\in [-2\pi ,2\pi ]$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (B)

$\frac{\pi }{4}$

We will write $1={\log }_{0.5}0.5[\because {\log }_{a}a=1]$
$\Rightarrow$ ${\log }_{0.5}\cos x={\log }_{0.5}0.5-$ ${\log }_{0.5}\sin x$
$\Rightarrow$ ${\log }_{0.5}\cos x$ + ${\log }_{0.5}\sin x$=${\log }_{0.5}0.5$

Using $\log a+\log b=\log \left(ab\right)$
$\Rightarrow$ ${\log }_{0.5}\left(\sin x\cos x\right)={\log }_{0.5}0.5$

$\Rightarrow$ $\cos x\sin x$ $=\frac{1}{2}$

$\Rightarrow$ $\sin 2x=1$
$\Rightarrow$ $2x=\frac{\pi }{2}$
$\Rightarrow$ $x=\frac{\pi }{4}$
Hence, option 'B' is correct.