Trigonometric Functions

We need to evaluate the expression . \textbf{Step 1: Understanding} : The function gives the angle whose cotangent is . We need to find the angle such that: The angle where is , because . Thus, \textbf{Step 2: Substituting into the original expression:} Now, substitute this value into the original expression: \textbf{Step 3: Simplifying the angle:} To simplify, add the angles: \textbf{Step 4: Finding :} From trigonometric values, we know that: Thus, the correct answer is (D) -1.

Let's break down the trigonometric expressions:

1. For : , since lies in the second quadrant where cosine is negative.
2. For : First, reduce the angle to an angle between 0° and 360°: Thus, , since .

So, we have:

The correct answer is (A) :

To convert radians to degrees, use the formula:


Given:

Now convert decimal degrees into degrees, minutes, and seconds:



So:

Correct Answer:

Let's simplify the expression step by step:

Let the expression be:

We use the identity:
Also, iteratively,

From trigonometric nested root identities,

This identity is derived using the recursive form of cosine angle reduction.

Correct Answer:

We are given the linear differential equation:

This is a first-order linear differential equation.
Let’s solve it using the integrating factor (IF) method.

Step 1: Identify the integrating factor (IF):

Step 2: Multiply the entire equation by the IF (x):

Step 3: Integrate both sides:

Step 4: Solve for :

Step 5: Compute :

Final Answer:

Given the expression:

We differentiate the expression with respect to using the sum rule, which allows us to differentiate each term separately.

For the first term, , the derivative is:

For the second term, , the derivative is:

Now, we can find by adding the derivatives of both terms:

To evaluate , we substitute into the equation:

Simplifying the trigonometric functions and evaluating the values:

Substituting the values gives:

After simplifying:

Thus, the value of at is 1, which corresponds to option (B) 1.