First slide

Trigonometric equations

Question

No of solutions of equation $\tan x + s e c x = 2 \cos x$ , $x \in [0, 2 π]$ is

Moderate

A

1
B

2
C

3
D

4

Solution

$\tan x + \sec x = 2 \cos x$
$\Rightarrow \frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2 \cos x \Rightarrow \sin x + 1 = 2 \cos^{2} x$
$\Rightarrow \sin x + 1 = 2 (1 - \sin^{2} x) \Rightarrow 2 \sin^{2} x + \sin x - 1 = 0$
$\Rightarrow 2 \sin^{2} x + 2 \sin x - \sin x - 1 = 0$
$\Rightarrow 2 \sin x (\sin x + 1) - 1 (\sin x + 1) = 0$
$\Rightarrow (2 \sin x - 1) (\sin x + 1) = 0$
$\Rightarrow \sin x = \frac{1}{2}$ or $\sin x = - 1$
$\Rightarrow \sin x = \sin \frac{π}{6}$ or $\sin x = \sin (- \frac{π}{2})$
$\Rightarrow x = n π + {(- 1)}^{n} \frac{π}{6}$ or $x = 2 n π - \frac{π}{2}$
$\Rightarrow f o r n = 0, x = \frac{π}{6}, x = \frac{- π}{2}$
$f o r n = 1, x = π - \frac{π}{6} = \frac{5 π}{6}, x = 2 π - \frac{π}{2} = \frac{3 π}{2}$
$\frac{3 π}{2} d o e s n o t s a t i s y t h e g i v e n e q u a t i o n$
$f o r n = 2, x = 2 π + \frac{π}{6}, x = 2 π - \frac{π}{2} = \frac{7 π}{2}$
$∴ x = \frac{π}{6}, \frac{5 π}{6} \in [0, 2 π] a r e t h e o n l y s o l u t i o n s$

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