Exam

NTA Abhyas

Year

2020

Subject

Mathematics

Chapter

Trigonometric Equations

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The number of roots of the equation tan x + sec x =2 cos x in [0,...

The number of roots of the equation tan x + sec x =2 cos x in $\left[0, 4\pi\right]$ is

Step-by-step solution

$tan x + sec x = 2cos x$
Multiplying by $cos x$ ,
$sin x + 1 = 2\cos^{2} x = 2 - 2\sin^{2} x$
$⇒ 2\sin^{2} x + sin x - 1 = 0 ⇒ sin x = - 1, \dfrac{1}{2}$

$sin x = \dfrac{1}{2}$ at
$x = \dfrac{π}{6}, \dfrac{5\pi}{6}$ in \left[0,2\pi\right]$<br />$sin x = \dfrac{1}{2} \to$four times in \left[0,4\pi\right]$
$sin x = - 1 ⇒ x = \dfrac{3\pi}{2}$ for which $\tan x and \sec x$ are not defined
$⇒ sin x = - 1$ is not possible
$⇒$ Number of roots in \left[0,4\pi\right]$is$4 .