Question

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

• A. 2
• B. 3
• C. 0
• D. 1

Hint:

Always check for values that make the original terms (like $\tan x$ or $\sec x$) undefined.

Answer 1

The Correct Option is A

Step 1: Concept
Express the equation in terms of $\sin x$ and $\cos x$.
Step 2: Analysis
$\frac{\sin x + 1}{\cos x} = 2 \cos x \Rightarrow 1 + \sin x = 2 \cos^{2} x = 2(1 - \sin^{2} x)$.
$2 \sin^{2} x + \sin x - 1 = 0 \Rightarrow (2 \sin x - 1)(\sin x + 1) = 0$.
$\sin x = 1/2$ or $\sin x = -1$.
Step 3: Conclusion
For $\sin x = 1/2$, $x = \pi/6, 5\pi/6$. $\sin x = -1$ makes $\cos x = 0$, which makes $\tan x$ undefined. Thus, only 2 valid solutions exist.
Final Answer: (A)