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Number of solutions of the equation $\tan x+\sec x=2\cos x$ lying in the interval $[0,2\pi]$ is

$\begin{array}{1 1}(A)\;0&(B)\;1\\(C)\;2&(D)\;3\end{array} $

1 Answer

Given :
$\tan x+\sec x =2\cos x$
$\large\frac{\sin x}{\cos x}+\frac{1}{\cos x}$$=2\cos x$
$(\sin x+1)=2\cos^2x$
$\sin x+1=2-2\sin^2x$
$2\sin^2x+\sin x-1=0$
$2\sin^2x+2\sin x-\sin x-1=0$
$2\sin x(\sin x+1)-1(\sin x+1)=0$
$(2\sin x-1)(\sin x+1)=0$
If $2\sin x-1=0$
$\sin x=\large\frac{1}{2}$
If $\sin x+1=0$
$\sin x=-1$
Hence (C) is the correct option.
answered Apr 21, 2014 by sreemathi.v

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