Trigonometric Identities

To find the height of the jet plane flying at a constant height, we will use trigonometry. The problem is set up with two angles of elevation, and the plane travels a certain distance in the horizontal direction.

1. First, convert the speed of the jet from km/h to m/s:

1. Calculate the horizontal distance covered by the plane in 20 seconds at this speed:

1. Consider the point A from which the angle of elevation is observed. Initially, the angle is . Let the height of the plane be and the initial horizontal distance from point A to the point right under the plane be . From the angle of elevation:

1. After 20 seconds, the angle changes to , and the horizontal distance is . Using this:

1. Since both expressions represent the height , set them equal to solve for :

Multiply through by :

1. Substitute back into the equation for :

Therefore, the height of the jet plane is the correct option: .

To solve this problem, we need to evaluate the expression and simplify the terms involved in the given inverse cotangent expressions. The expression is:

First, recognize that simplifies using the identity . Thus:

1. and .

For the first term:

Using the identity , the expression becomes:

For the second term:

In this case, evaluate using the complementary angle identity, which suggests manipulating the cotangent identity appropriately. Since:

Adjustment yields:

1. becomes

But this requires careful simplification and conceptual understanding of cotangent addition:

1. evaluates to the adjusted angle associated with the angles sum.

Ultimately, upon recognizing both co-tangents and trigonometric simplifications, the expression simplifies using angle subtraction identities:

1. after evaluating through symmetry and exact measures.

Thus, the value of the expression is . This analysis confirms the correct answer is:

This detailed step-by-step breakdown leverages trigonometric identities and simplifications, applied correctly, to obtain the solution. Understanding angle transformations in trigonometry is crucial for queries involving inverse trigonometric functions.

To solve this problem, we need to simplify the expression given the range .

1. We recognize that the expression inside the inverse cosine function resembles the application of a known trigonometric identity:

Let and .

1. Notice that . Therefore, the expression can be transformed into a standard cosine formula:

.

1. We need to find the angle , where:
2. .
3. Thus, we have .
4. Since the angles lie in the domain where cosine is non-increasing, .
5. Subtracting the angle gives us: .

This matches with the given option:

.

Hence, the correct answer is
Option: .

Step 1: Simplifying the given expression.
Group the terms: Using identities, Thus, the expression becomes Step 2: Using sine–cosine product identity.
Using After simplification, the expression reduces to Step 3: Finding .
Given Since , lies in the third quadrant, Using a right triangle, Step 4: Final evaluation.
Taking magnitude as required,