Question

Find the no. of roots of the equation $\tan x+\sec x=2$
$cosx$ in the interval $[0,2\pi ]$.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

$3$

$\tan x+\sec x=2\cos x$

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

$\Rightarrow \sin x+1=2{\cos }^{2}x$

$\Rightarrow \sin x+1=2(1-{\sin }^{2}x)$

$\Rightarrow \sin x+1=2-2{\sin }^{2}x$

$\Rightarrow 2{\sin }^{2}x+\sin x-1=0$

$\Rightarrow 2{\sin }^{2}x+2\sin x-\sin x-1=0$

$\Rightarrow 2\sin x(\sin x+1)-1(\sin x+1)=0$

$\Rightarrow (2\sin x-1)(\sin x+1)=0$

$\Rightarrow \sin x=\frac{1}{2}$; $\sin x=-1$

$x=\frac{3\pi }{2};\frac{\pi }{6};\frac{5\pi }{6}$.