Question

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

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is C 2

$tanx+secx=2cosx$

$sinx+1=2co{s}^{2}x=2-2si{n}^{2}x$

$2si{n}^{2}x+sinx-1=0$

$(2sinx-1)(sinx+1)=0$

but $sinx=-1\Rightarrow x=\frac{3\pi }{2}$, which does not satisfy the given equation.

$sinx=\frac{1}{2}=sin\frac{\pi }{6}$

therefore general solution is,
$x=n\pi +(-1{)}^n.\frac{\pi }{6}$

$x=......,\frac{\pi }{6},\frac{5\pi }{6},......$

hence number of solution in the given interval is 2.