Trigonometry

Step 1: We are given , and we need to simplify the expression.

Step 2: First, simplify by reducing it within the range of the sine inverse function, which is . To do this, observe that: Since is periodic with period , we have:

Step 3: Now, lies in the fourth quadrant. In the fourth quadrant, .

Step 4: Therefore, we have:

Step 5: The angle whose sine is in the range is .

Thus,

Step 1: Begin by evaluating each term separately.

Step 2: For , we know that: since the tangent of is .

Step 3: Next, evaluate . The value of corresponds to the angle such that .
Since , we have , and the angle that satisfies this condition is (since ).

Step 4: Now, evaluate . The angle whose sine is is , since .

Step 5: Now, add all the results together:

Step 6: Simplify the expression: so the expression becomes: Thus, the final value is: