Question

The number of solutions of the equation $\frac{\sec x}{1-\cos x}=\frac{1}{1-\cos x}$ in $[0,2\pi ]$ is equal to?

• A. $3$
• B. $2$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

We have equation

$\frac{\sec x}{1-\cos x}=\frac{1}{1-\cos x}$ in $x\in [0,2\pi ]$

$(1-\cos x)\sec x=(1-\cos x)$

$(1-\cos x)\sec x-(1-\cos x)=0$

$(1-\cos x)(\sec x-1)=0$

$1-\cos x=0$ or $\sec x-1=0$

$\cos x=1$ or $\sec x=1$

$x={\cos }^{-1}\left(1\right)$ or $x={\sec }^{-1}\left(1\right)$

$x=0$. or $2\pi$ $x=0$

but in the original equation we can put $x=0$ or $x=2\pi$ as It goes to undetermine value (as infinity)

So, The number of solution is $0$