This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: Problem 1: Prove + = Step 1: Start with the Left Hand Side (LHS). LHS = + Step 2: Express as ( )/( ). LHS = ( )/( ) · + LHS = (^2 )/( ) + Step 3: Find a common denominator and combine the terms. LHS = (^2 + ^2 )/( ) Step 4: Use the Pythagorean identity ^2 + ^2 = 1. LHS = (1)/( ) Step 5: Express in terms of . LHS = Since LHS = RHS, the identity is proven. Proven --- Problem 2: Prove (1)/( ) + = (1)/( ) Step 1: Start with the Left Hand Side (LHS). LHS = (1)/( ) + Step 2: Express (1)/( ) as ( )/( ) and as ( )/( ). LHS = ( )/( ) + ( )/( ) Step 3: Find a common denominator, which is . LHS = (^2 )/( ) + (^2 )/( ) Step 4: Combine the terms. LHS = (^2 + ^2 )/( ) Step 5: Use the Pythagorean identity ^2 + ^2 = 1. LHS = (1)/( ) Since LHS = RHS, the identity is proven. Proven --- Problem 3: If A = (3)/(5), prove that A + (1)/( A) = 2 or -2. Step 1: Given A = (3)/(5). Use the identity ^2 A + ^2 A = 1 to find A. ((3)/(5))^2 + ^2 A = 1 (9)/(25) + ^2 A = 1 ^2 A = 1 - (9)/(25) = (16)/(25) A = ± sqrt((16)/(25)) = ± (4)/(5) Step 2: Consider two cases for A. Case 1: A = (4)/(5) (A is in Quadrant I) Calculate A: A = ( A)/( A) = (3/5)/(4/5) = (3)/(4) Substitute values into the expression A + (1)/( A): A + (1)/( A) = (3)/(4) + (1)/(4/5) = (3)/(4) + (5)/(4) = (8)/(4) = 2 Case 2: A = -(4)/(5) (A is in Quadrant II) Calculate A: A = ( A)/( A) = (3/5)/(-4/5) = -(3)/(4) Substitute values into the expression A + (1)/( A): A + (1)/( A) = -(3)/(4) + (1)/(-4/5) = -(3)/(4) - (5)/(4) = (-8)/(4) = -2 Thus, A + (1)/( A) can be 2 or -2. Proven --- **Problem 4: If is in the fourth quadrant and = (5)/(13), find the value of $13 + 5 5