Question

The number of values of $x\in [0,n\pi ],n\in I$ that satisfy $lo{g}_{|sinx|}(1+cosx)=2$ is

• A. 0
• B. n
• C. 2n
• D. None of these

Answer 1

The correct option is A

0

The equation is meaningful if $|sinx|\ne 0or1$ and $1+cosx\ne 0$

so $x\ne k\pi ,k=0,1,\dots n,andx\ne (2k+1)\frac{\pi }{2},k=0,...n-1$.

$lo{g}_{|sinx|}(1+cosx)=2$
$\iff 1+cosx=|sinx{|}^2=si{n}^{2}x=1-co{s}^{2}x\Rightarrow (1+cosx)\left(cosx\right)=0\iff cosx=0orcosx=-1\Rightarrow cosx=0\Rightarrow x=(2k+1)\frac{\pi }{2}.$
$\Rightarrow cosx=-1\Rightarrow x=k\pi .$
So there is no $x$ which satisfy the given equation.