When n is an odd natural number other than $1,$ then the value of x is
This question has multiple correct options
A. $- \dfrac{\pi }{2}$
B. $0$
C. $\pi $
D. $3\pi $

Hint: To check the given multiple choices we will place them one by one and place in the cosine and sine function and will find the value of “x” and then will find the correct options. Also use the trigonometric table for reference values.

Complete step by step answer:
Start checking the given options one by one.
Option A
${\cos ^n}\left( {\dfrac{{ - \pi }}{2}} \right) = 0$
Similarly, ${\sin ^n}\left( {\dfrac{{ - \pi }}{2}} \right) = ( - 1)$
Since we are given that “n” is an odd natural number.
Therefore, option A is the correct answer. …. (A)
Option B
${\cos ^n}\left( 0 \right) = 1$
Similarly, ${\sin ^n}\left( 0 \right) = 0$
Since we are given that “n” is an odd natural number.
Therefore, option B is the correct answer. …. (B)
Option C
${\cos ^n}\left( { - \pi } \right) = ( - 1)$
Similarly, ${\sin ^n}\left( \pi \right) = 0$
Since we are given that “n” is an odd natural number.
Therefore, option C is not the correct answer. …. (C)
Option D
${\cos ^n}\left( {3\pi } \right) = ( - 1)$
Similarly, ${\sin ^n}\left( {3\pi } \right) = 0$
Since we are given that “n” is an odd natural number.
Therefore, option D is not the correct answer. …. (D)

From the given multiple choices, the options A and B are the correct answers.

Additional Information:
Remember the All STC rule, it is also known as ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ $ ).

Note:Be careful in identifying any degree of measure in the one of the four quadrants and also refer to sine and cosine even and odd functions. Remember the trigonometric table for the reference values for different angles for sine, cosine and tangent functions for direct substitution for the accurate and the efficient solution.