Trigonometric Identities

To solve the problem, we need to simplify the given expression using trigonometric identities and the given value of .

1. Given information:
Expression to simplify:

2. Group terms involving and :
Group as:

3. Use cosine and sine addition formulas:
Recall:
So, and

4. Simplify the angles:
Therefore, the expression becomes:

5. Use even-odd properties of trig functions:
So,

6. Express in a single trigonometric function:

7. Find using the value of :
Given: Since , and . Let's find and :

8. Calculate and :
From , let: Using Pythagoras: So,

9. Use half-angle formulas:

10. Calculate :
Using sum formula:

11. Substitute back:

Final Answer:
The value of the given expression is: