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.
ScanSolve AI Answer
Hey T.Hossain, good to see you again. Here's the solution for problem 15: We need to prove that ( A)/(1 - A) + (1 - A)/( A) = 2cosec A. Step 1: Start with the Left Hand Side (LHS) of the equation. LHS = ( A)/(1 - A) + (1 - A)/( A) Step 2: Find a common denominator for the two fractions, which is (1 - A) A. LHS = ( A · A + (1 - A)(1 - A))/((1 - A) A) LHS = (^2 A + (1 - A)^2)/((1 - A) A) Step 3: Expand the term (1 - A)^2 in the numerator. Recall the algebraic identity (a-b)^2 = a^2 - 2ab + b^2. So, (1 - A)^2 = 1^2 - 2(1)( A) + ^2 A = 1 - 2 A + ^2 A. Substitute this back into the expression: LHS = (^2 A + 1 - 2 A + ^2 A)/((1 - A) A) Step 4: Rearrange the terms in the numerator and use the trigonometric identity ^2 A + ^2 A = 1. LHS = ((^2 A + ^2 A) + 1 - 2 A)/((1 - A) A) LHS = (1 + 1 - 2 A)/((1 - A) A) LHS = (2 - 2 A)/((1 - A) A) Step 5: Factor out 2 from the numerator. LHS = (2(1 - A))/((1 - A) A) Step 6: Cancel out the common term (1 - A) from the numerator and the denominator. LHS = (2)/( A) Step 7: Use the reciprocal identity cosec A = (1)/( A). LHS = 2 · (1)/( A) = 2cosec A This is equal to the Right Hand Side (RHS). Therefore, the identity is proven: ( A)/(1 - A) + (1 - A)/( A) = 2cosec A What's next? Send 'em! 📸