Question

If $0<x<\frac{\pi }{2}$ and $\cos x=\frac{3}{\sqrt{10}}$, then the value of ${\log }_{10}\sin x+{\log }_{10}\cos x+{\log }_{10}\tan x$ is equal to

• A. $0$
• B. $1$
• C. $-1$
• D. none of these

Answer 1

The correct option is C $-1$

Given that, $0<x<\frac{\pi }{2}$ and $\cos x=\frac{3}{\sqrt{10}}$
Let, $a={\log }_{10}\sin x+{\log }_{10}\cos x+{\log }_{10}\tan x$
$\Rightarrow a={\log }_{10}\left(\sin x\cos x\tan x\right)[\because \log a+\log b+\log c=\log (abc\left)\right]$

$={\log }_{10}{\sin }^{2}x$

$={\log }_{10}(1-{\cos }^{2}x)$
$\Rightarrow a={\log }_{10}(1-{\left(\frac{3}{\sqrt{10}}\right)}^2)={\log }_{10}{10}^{-1}=-1{\log }_{10}10=-1,[\because {\log }_{a}a=1]$
Ans: C