Trigonometric Functions

First, simplify the given equation by expanding the terms: Using : Now, focus on finding . Start with expressing using the angle-sum formula and Pythagorean identity: In the third quadrant, both and are negative, and their sum squared would be: For the third quadrant where , is negative. Assuming for calculation: Cubing this result:

To solve for , we utilize the double-angle and trigonometric transformation formulas. First, let's establish some basic trigonometric relationships and assumptions based on the given equations.

Step 1: Simplify the given expressions and find relationships. Given: We need to express and in terms of each other or find a common angle expression. For simplification purposes, assume some angle identities or derive them based on given equations.

Step 2: Express in terms of cosine and sine transformations. Using the given, Knowing from Pythagorean identity, Combining the above with , equate and solve these using assumed angles or calculate using trigonometric tables.

Step 3: Calculate . Transform using double-angle identities: Assuming simplifies to based on possible angle simplifications.