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
Alright Rajnish — let's do this. Here's the solution for question 25 (a) and (b): 25 (a) Prove that sqrt((1+ A)/(1- A)) = A + A Step 1: Start with the Left Hand Side (LHS). LHS = sqrt((1+ A)/(1- A)) Step 2: Multiply the numerator and denominator inside the square root by the conjugate of the denominator, (1+ A). LHS = sqrt((1+ A)/(1- A) × (1+ A)/(1+ A)) LHS = sqrt(((1+ A)^2)/(1^2 - ^2 A)) Step 3: Use the identity 1 - ^2 A = ^2 A. LHS = sqrt(((1+ A)^2)/(^2 A)) Step 4: Take the square root of the numerator and the denominator. LHS = sqrt((1+ A)^2)sqrt(^2 A) LHS = (1+ A)/( A) Step 5: Separate the terms. LHS = (1)/( A) + ( A)/( A) Step 6: Express in terms of A and A. LHS = A + A This is equal to the Right Hand Side (RHS). LHS = RHS Hence, proved. Proved 25 (b) Evaluate 3 ^2 30^ - 6 cosec^2 30^^2 60^ Step 1: Recall the values of the trigonometric functions. 30^ = sqrt(3)2 cosec 30^ = (1)/( 30^) = (1)/(1/2) = 2 60^ = sqrt(3) Step 2: Substitute these values into the expression. 3 (sqrt(3)2)^2 - 6 (2)^2(sqrt(3))^2 Step 3: Calculate the squares. (3 (3)/(4)) - 6 (4)3 (9)/(4) - 243 Step 4: Simplify the numerator. (9)/(4) - (96)/(4)3 (-87)/(4)3 Step 5: Perform the division. -(87)/(4) × (1)/(3) -(87)/(12) Step 6: Simplify the fraction by dividing the numerator and denominator by 3. -(29)/(4) The evaluated value is -(29)/(4). Send me the next one 📸