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  4. Number of solutions of the equation (2 operatornamecosec x-1+( operatornamecosec x-1=1 in (-k π, k π) is 16 , then possible value of ' k ' is

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Q. Number of solutions of the equation $(2 \operatorname{cosec} x-1+(\operatorname{cosec} x-1=1$ in $(-k \pi, k \pi)$ is 16 , then possible value of ' $k$ ' is

NTA AbhyasNTA Abhyas 2022

Solution:

$(2 \operatorname{cosec} x-1+(\operatorname{cosec} x-1=1$
cube both sides
$(3 \operatorname{cosec} x-2)+3(2 \operatorname{cosec} x-1(\operatorname{cosec} x-1$ $=1$
$(2 \operatorname{cosec} x-1(\operatorname{cosec} x-1=(1-\operatorname{cosec} x)$
$(2 \operatorname{cosec} x-1=-(\operatorname{cosec} x-1$
$\Rightarrow \operatorname{cosec} x=0$ (neglect)
or
$\operatorname{cosec} x=1 \Rightarrow \sin x=1, \quad x=2 n \pi+\frac{\pi}{2}$
$\Rightarrow 1$ solution in $(0,2 \pi)$
$\Rightarrow k$ solutions in $(-k \pi, k \pi) \Rightarrow k=16$