Fraction

We are tasked with solving the equation:

and finding the value of .

Step 1: Analyze the numerator and denominator.

The numerator is the sum of the first terms of an arithmetic progression (AP) with first term and common difference . The denominator is the sum of the first 8 terms of another AP with first term and common difference .

Step 2: Use the formula for the sum of an AP.

The sum of the first terms of an AP is given by:

Step 3: Compute the denominator.

For the denominator, , , and :

Step 4: Express the numerator.

For the numerator, , , and :

Step 5: Set up the equation.

Substitute the expressions for the numerator and denominator into the given equation:

Simplify:

Multiply through by 200:

Step 6: Solve the quadratic equation.

Rearrange into standard quadratic form:

Factorize:

Thus, or . Since must be positive, we have:

Final Answer: The value of is .

Step 1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of is . Multiply both numerator and denominator by :

Step 2: Simplify the denominator using the difference of squares formula.

The denominator becomes:

Using the Pythagorean identity , the expression becomes:

Step 3: Simplify the fraction.

Cancel one from the numerator and denominator:

Final Answer: The simplified value of is , which corresponds to option .

We are tasked with finding the value of given that:

Step 1: Substitute the expressions for and into .

First, compute and :

Substitute these into :

Simplify:

Step 2: Factor out .

Step 3: Use the trigonometric identity .

From the Pythagorean identity:

Substitute this into the expression:

Simplify:

Final Answer: The value of is , which corresponds to option .

We are given the expression: To simplify this expression, multiply both the numerator and the denominator by : Using the identity , we simplify the denominator: Thus, the expression becomes: Therefore, the simplified expression is:

The correct option is (C):