Question

If $\cot \theta =3x-\frac{1}{12x}$, then $\csc \theta -\cot \theta$ is equal to

• A. $6x$
• B. $-\frac{1}{6x}$
• C. $\frac{1}{6x}$ or $-6x$
• D. $6x$ or $-\frac{1}{6x}$

Answer 1

Answer: (C)

$\frac{1}{6x}$ or $-6x$

Given: $\cot \theta =3x-\frac{1}{12x}$

We know ${\csc }^2\theta -{\cot }^2\theta =1$

$\Rightarrow {\csc }^2\theta =1+{\cot }^2\theta$

$\Rightarrow {\csc }^2\theta =1+(3x-\frac{1}{12x}{)}^2$

$\Rightarrow {\csc }^2\theta =1+9x^2+\frac{1}{144x^2}-\frac{1}{2}$

$\Rightarrow {\csc }^2\theta =9x^2+\frac{1}{144x^2}+\frac{1}{2}$

$\Rightarrow {\csc }^2\theta =(3x+\frac{1}{12x}{)}^2$

$\Rightarrow \csc \theta =\pm (3x+\frac{1}{12x})$

If $\csc \theta =3x+\frac{1}{12x}$

Then $\csc \theta -\cot \theta =3x+\frac{1}{12x}-(3x-\frac{1}{12x})$

$=\frac{1}{6x}$

If $\csc \theta =-(3x+\frac{1}{12x})=-3x-\frac{1}{12x}$

Then $\csc \theta -\cot \theta =-3x-\frac{1}{12x}-(3x-\frac{1}{12x})$

$=-6x$

Hence $\csc \theta -\cot \theta =-6x$ or $\frac{1}{6x}$