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  4. If log 2(3 sin x)- log 2( cos x)- log 2(1- tan x)- log 2(1+ tan x)=1, then tan x is equal to

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Q. If $\log _2(3 \sin x)-\log _2(\cos x)-\log _2(1-\tan x)-\log _2(1+\tan x)=1$, then $\tan x$ is equal to

Continuity and Differentiability

Solution:

We have
$ \log _2(3 \sin x)-\log _2(\cos x)-\log _2(1-\tan x)-\log _2(1+\tan x)=1 $
$\Rightarrow \log _2\left(\frac{3 \sin x}{\cos x(1-\tan x)(1+\tan x)}\right)=1 \Rightarrow \log _2\left(\frac{3 \tan x}{1-\tan ^2 x}\right)=1$
$\Rightarrow 3 \tan x=2\left(1-\tan ^2 x\right) \Rightarrow 2 \tan ^2 x+3 \tan x-2=0$
$\Rightarrow 2 \tan ^2 x+4 \tan x-\tan x-2=0 (2 \tan x-1)(\tan x+2)=0 $
$\Rightarrow \tan x=\frac{1}{2} \text { or } \tan x=-2(\text { rejected, think!) }$