Trigonometry

Given ,

differentiate both sides w.r.t :

Simplify:

Substitute :

Thus:

Multiply both sides by :

The given differential equation is:

Rearranging terms: This is a linear differential equation of the form: where and .

Step 1: Solve the homogeneous equation. The associated homogeneous equation is:
Separating variables: Integrating both sides: where is the constant of integration. Simplify:

Step 2: Solve the non-homogeneous equation using an integrating factor. The integrating factor (IF) is:
Multiply through the original equation by :

Simplify: Integrating both sides: Using the initial condition , , we find .
Solve further as needed.

The particular solution will depend on further simplification or numerical methods to compute the integral.

The direction cosines of a line satisfy the relation:
where are the direction cosines.

Here:

Substitute into the equation:

Simplify: Thus:

The correct answer is (D) .

Step 1: Simplify
Let . Then: Using the identity :
Step 2: Simplify
Let . Then: Using the identity :
Step 3: Combine the results
Simplify:
Conclusion: The value is .

The given function is:
Take the natural logarithm on both sides to simplify the power:

Differentiate both sides with respect to :

Simplify:

Multiply through by :

Simplify further:

Final answer:

The given function is:

Differentiate w.r.t. : At : Since :

Therefore, the correct answer is (C) .

To solve the problem, we need to evaluate the expression:

1. Understand the Inverse Cotangent:
We start by evaluating .
Let
Then,
We know that , so:

2. Plug Back into the Original Expression:
Now substitute this value into the original expression:

3. Evaluate the Cosine:

Final Answer:
The value of is .