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Hint: Here, we will use when \[\alpha \] is an angle in degree measure then the relation between circular and degree measure is \[{\alpha ^C} = \dfrac{\pi }{{180}} \times \alpha \].

Apply this relation, and then use the given conditions to find the required value.

Given that the difference of the two angles in degree measure is 1 and their sum in radian measure is also 1.

Let us assume that the two angles are \[X\] and \[Y\] in degree measure.

We know that if \[\alpha \] is an angle in degree measure then the relation between circular and degree measure is \[{\alpha ^C} = \dfrac{\pi }{{180}} \times \alpha \].

First, we will find the equation representing the difference of the two angles in degree measure is 1.

\[X - Y = 1\]

But since the difference of the two angles is in degree measure, so we can write \[\dfrac{\pi }{{180}} = 1\].

Using this value of 1 in the above equation, we get

\[X - Y = \dfrac{\pi }{{180}}{\text{ ......}}\left( 1 \right)\]

We will now find the equation representing the sum of the two angles in radian measure is 1.

\[X + Y = 1{\text{ ......}}\left( 2 \right)\]

Subtracting the above equation by \[Y\] on each of the sides, we get

\[

\Rightarrow X + Y - Y = 1 - Y \\

\Rightarrow X = 1 - Y{\text{ .......}}\left( 3 \right) \\

\]

Substituting this value of \[X\] in the equation \[\left( 1 \right)\], we get

\[

\Rightarrow \left( {1 - Y} \right) - Y = \dfrac{\pi }{{180}} \\

\Rightarrow 1 - 2Y = \dfrac{\pi }{{180}} \\

\]

Subtracting the above equation by 1 on each of the sides, we get

\[

\Rightarrow 1 - 2Y - 1 = \dfrac{\pi }{{180}} - 1 \\

\Rightarrow - 2Y = \dfrac{\pi }{{180}} - 1 \\

\]

Dividing the above equation by \[ - 2\] on each of the sides, we get

\[

\Rightarrow \dfrac{{ - 2Y}}{{ - 2}} = \dfrac{1}{{ - 2}}\left( {\dfrac{\pi }{{180}} - 1} \right) \\

\Rightarrow Y = - \dfrac{\pi }{{360}} + \dfrac{1}{2} \\

\Rightarrow Y = \dfrac{1}{2} - \dfrac{\pi }{{360}} \\

\]

Substituting this value of \[Y\] in the equation \[\left( 3 \right)\], we get

\[

\Rightarrow X = 1 - \left( {\dfrac{1}{2} - \dfrac{\pi }{{360}}} \right) \\

\Rightarrow X = 1 - \dfrac{1}{2} + \dfrac{\pi }{{180}} \\

\Rightarrow X = \dfrac{1}{2} + \dfrac{\pi }{{360}} \\

\]

Therefore, the angles in circular measure are \[\dfrac{1}{2} + \dfrac{\pi }{{360}}\] and \[\dfrac{1}{2} - \dfrac{\pi }{{360}}\].

Hence, the option A is correct.

Note: In solving these types of questions, you should be familiar with the concept of circular measure and degree measures. Then use the given conditions and values given in the question, to find the required values. Also, we are supposed to write the values properly to avoid any miscalculation.

Apply this relation, and then use the given conditions to find the required value.

__Complete step-by-step solution:__Given that the difference of the two angles in degree measure is 1 and their sum in radian measure is also 1.

Let us assume that the two angles are \[X\] and \[Y\] in degree measure.

We know that if \[\alpha \] is an angle in degree measure then the relation between circular and degree measure is \[{\alpha ^C} = \dfrac{\pi }{{180}} \times \alpha \].

First, we will find the equation representing the difference of the two angles in degree measure is 1.

\[X - Y = 1\]

But since the difference of the two angles is in degree measure, so we can write \[\dfrac{\pi }{{180}} = 1\].

Using this value of 1 in the above equation, we get

\[X - Y = \dfrac{\pi }{{180}}{\text{ ......}}\left( 1 \right)\]

We will now find the equation representing the sum of the two angles in radian measure is 1.

\[X + Y = 1{\text{ ......}}\left( 2 \right)\]

Subtracting the above equation by \[Y\] on each of the sides, we get

\[

\Rightarrow X + Y - Y = 1 - Y \\

\Rightarrow X = 1 - Y{\text{ .......}}\left( 3 \right) \\

\]

Substituting this value of \[X\] in the equation \[\left( 1 \right)\], we get

\[

\Rightarrow \left( {1 - Y} \right) - Y = \dfrac{\pi }{{180}} \\

\Rightarrow 1 - 2Y = \dfrac{\pi }{{180}} \\

\]

Subtracting the above equation by 1 on each of the sides, we get

\[

\Rightarrow 1 - 2Y - 1 = \dfrac{\pi }{{180}} - 1 \\

\Rightarrow - 2Y = \dfrac{\pi }{{180}} - 1 \\

\]

Dividing the above equation by \[ - 2\] on each of the sides, we get

\[

\Rightarrow \dfrac{{ - 2Y}}{{ - 2}} = \dfrac{1}{{ - 2}}\left( {\dfrac{\pi }{{180}} - 1} \right) \\

\Rightarrow Y = - \dfrac{\pi }{{360}} + \dfrac{1}{2} \\

\Rightarrow Y = \dfrac{1}{2} - \dfrac{\pi }{{360}} \\

\]

Substituting this value of \[Y\] in the equation \[\left( 3 \right)\], we get

\[

\Rightarrow X = 1 - \left( {\dfrac{1}{2} - \dfrac{\pi }{{360}}} \right) \\

\Rightarrow X = 1 - \dfrac{1}{2} + \dfrac{\pi }{{180}} \\

\Rightarrow X = \dfrac{1}{2} + \dfrac{\pi }{{360}} \\

\]

Therefore, the angles in circular measure are \[\dfrac{1}{2} + \dfrac{\pi }{{360}}\] and \[\dfrac{1}{2} - \dfrac{\pi }{{360}}\].

Hence, the option A is correct.

Note: In solving these types of questions, you should be familiar with the concept of circular measure and degree measures. Then use the given conditions and values given in the question, to find the required values. Also, we are supposed to write the values properly to avoid any miscalculation.

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