Question

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

• A. Zero
• B. One
• C. Two
• D. Three

Answer 1

The correct option is D Three

$\tan x+\sec x=2\cos x$

$\Rightarrow$$\frac{\sin x}{\cos x}+\frac{1}{\cos x}=2\cos x$

$\Rightarrow$$\frac{(\sin x+1)}{\cos x}=2\cos x$

$\Rightarrow$$1+\sin x=2{\left(\cos x\right)}^2$

$\Rightarrow$$1+\sin x=2(1-{\left(\sin x\right)}^2)$

$\Rightarrow$$1+\sin x-2(1+\sin x)(1-\sin x)=0$

$\Rightarrow$$(1+\sin x)(1-2(1-\sin x\left)\right)=0$

$\Rightarrow$$(1+\sin x)(2\sin x-1)=0$

$\Rightarrow$$\sin x=1,\sin x=\frac{1}{2}$

Since, $x$ is between $[0,2\pi ]$,

The solutions are $x=\frac{3\pi }{2},x=\frac{\pi }{6},x=\frac{5\pi }{6}$.

Therefore there are 3 solutions.