Answer

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Hint- In order to solve such a type of question directly use the trigonometric identity. First find out the quadrant in which the given angle lies and then try to convert it in terms of some angle with known trigonometric value.

Complete step-by-step solution -

Given: ${\text{cosec}}\left( { - {{90}^0}} \right)$

As we know that the position of angles in the quadrant system remains the same after every ${360^0}$ .

So the angle $ - {90^0}$ coincides with ${270^0}$ .

So the angle remains between the 3rd and 4th quadrant.

As we know that in the 3rd and 4th quadrant the value of ${\text{sin\& cosec}}$ are negative.

So we have the formula ${\text{cosec}}\left( { - \theta } \right) = - {\text{cosec}}\left( \theta \right)$

Using the above formula we have

\[{\text{cosec}}\left( { - {{90}^0}} \right) = - {\text{cosec}}\left( {{{90}^0}} \right)\]

As we know the value of ${\text{cosec}}\left( {{{90}^0}} \right) = 1$

$ \Rightarrow {\text{cosec}}\left( { - {{90}^0}} \right) = - {\text{cosec}}\left( {{{90}^0}} \right) = - \left( 1 \right)$

Hence, the value of ${\text{cosec}}\left( { - {{90}^0}} \right) = - 1$

So, option B is the correct option.

Note- In order to solve such a question where the exact value of the trigonometric term is not known try to bring it in the form of some known angles by the help of trigonometric identities. Also start with finding the quadrant of the angle in order to find the sign of final answer.

Complete step-by-step solution -

Given: ${\text{cosec}}\left( { - {{90}^0}} \right)$

As we know that the position of angles in the quadrant system remains the same after every ${360^0}$ .

So the angle $ - {90^0}$ coincides with ${270^0}$ .

So the angle remains between the 3rd and 4th quadrant.

As we know that in the 3rd and 4th quadrant the value of ${\text{sin\& cosec}}$ are negative.

So we have the formula ${\text{cosec}}\left( { - \theta } \right) = - {\text{cosec}}\left( \theta \right)$

Using the above formula we have

\[{\text{cosec}}\left( { - {{90}^0}} \right) = - {\text{cosec}}\left( {{{90}^0}} \right)\]

As we know the value of ${\text{cosec}}\left( {{{90}^0}} \right) = 1$

$ \Rightarrow {\text{cosec}}\left( { - {{90}^0}} \right) = - {\text{cosec}}\left( {{{90}^0}} \right) = - \left( 1 \right)$

Hence, the value of ${\text{cosec}}\left( { - {{90}^0}} \right) = - 1$

So, option B is the correct option.

Note- In order to solve such a question where the exact value of the trigonometric term is not known try to bring it in the form of some known angles by the help of trigonometric identities. Also start with finding the quadrant of the angle in order to find the sign of final answer.

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