Question

Number of solutions of the equation$\mathbf{\tan}\mathbf{x}\mathbf{+}\mathbf{\sec}\mathbf{x}\mathbf{= 2}\mathbf{\cos}\mathbf{x}\mathbf{,}$ lying in the interval $\lbrack 0,2\pi\rbrack$ is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

2

Answer 2

$\frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2\cos x \Rightarrow \sin x + 1 = 2\cos^{2}x \Rightarrow 2\sin^{2}x + \sin x - 1 = 0$ $\Rightarrow$ $\left\lbrack 2\sin x - 1 \right\rbrack\left\lbrack \sin x + 1 \right\rbrack = 0$

So, $\sin x = - 1$ or $\sin x = \frac{1}{2} \Rightarrow x = \frac{3\pi}{2}$ or $x = \frac{\pi}{6},\frac{5\pi}{6}$but $\frac{3\pi}{2}$does not satisfy the equation, So total number of solutions$= 2.$