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Q.

If $cotθ = 3 x - \frac{1}{12 x}$ , then $cscθ - cotθ$ is equal to ?

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a

6 x

b

- \frac{1}{6 x}

c

\frac{1}{6 x}

or

- 6 x

d

6 x

or

- \frac{1}{6 x}

answer is C.

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Detailed Solution

Given that,

cotθ = 3 x - \frac{1}{12 x}

.
We know the trigonometric identity,

\csc^{2} θ - \cot^{2} θ = 1

\Rightarrow \csc^{2} θ = 1 + \cot^{2} θ

Substituting the given value

cotθ = 3 x - \frac{1}{12 x}

in the above equation, we get

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

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

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

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

\Rightarrow cscθ = \pm (3 x + \frac{1}{12 x})

If

cscθ = 3 x + \frac{1}{12 x}

,
then

cscθ - cotθ

= 3 x + \frac{1}{12 x} - (3 x - \frac{1}{12 x})

= \frac{1}{6 x}

If

cscθ = - (3 x + \frac{1}{12 x}) = - 3 x - \frac{1}{12 x}

,
then

cscθ - cotθ

= - 3 x - \frac{1}{12 x} - (3 x - \frac{1}{12 x})

= - 6 x

Therefore,

cscθ - cotθ = - 6 x

or

\frac{1}{6 x}

.
Hence, the correct option is (3).

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