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 proofs for the trigonometric identities: a) Prove: ( x)/(1+ x) + ( x)/(1- x) = (2 x)/(-^2 x) Assumption: Due to the image clarity, we assume the Right Hand Side (RHS) denominator is 1-^2 x, which makes the identity provable. The identity will be proven as ( x)/(1+ x) + ( x)/(1- x) = (2 x)/(1-^2 x). Step 1: Start with the Left Hand Side (LHS) and factor out x. LHS = x ( (1)/(1+ x) + (1)/(1- x) ) Step 2: Combine the fractions inside the parenthesis. LHS = x ( ((1- x) + (1+ x))/((1+ x)(1- x)) ) Step 3: Simplify the numerator and multiply the denominators. LHS = x ( (2)/(1-^2 x) ) Step 4: Multiply to get the final expression. LHS = (2 x)/(1-^2 x) This matches the assumed RHS. Therefore, LHS = RHS. b) Prove: ( x)/(1+ x) - (1+ x)/( x) = 2 x Assumption: Due to the image clarity, we assume the operation between the two fractions is addition, not subtraction, which makes the identity provable. The identity will be proven as ( x)/(1+ x) + (1+ x)/( x) = 2 x. Step 1: Start with the LHS and find a common denominator to combine the fractions. LHS = ( x · x + (1+ x)(1+ x))/((1+ x) x) Step 2: Expand the numerator and use the identity ^2 x + ^2 x = 1. LHS = (^2 x + (1+2 x+^2 x))/((1+ x) x) LHS = ((^2 x + ^2 x) + 1 + 2 x)/((1+ x) x) = (1 + 1 + 2 x)/((1+ x) x) Step 3: Simplify the numerator by factoring out 2. LHS = (2 + 2 x)/((1+ x) x) = (2(1+ x))/((1+ x) x) Step 4: Cancel out the common term (1+ x). LHS = (2)/( x) Step 5: Use the identity (1)/( x) = x. LHS = 2 x This matches the RHS. Therefore, LHS = RHS. c) Prove: (^2 x - 1)/(^2 x + 1) = 1 - 2^2 x Step 1: Start with the LHS and express ^2 x in terms of ^2 x and ^2 x. LHS = (^2 x)/(^2 x) - 1(^2 x)/(^2 x) + 1 Step 2: Combine the terms in the numerator and denominator. LHS = (^2 x - ^2 x)/(^2 x)(^2 x + ^2 x)/(^2 x) Step 3: Simplify by canceling ^2 x from the numerator and denominator. LHS = (^2 x - ^2 x)/(^2 x + ^2 x) Step 4: Use the identity ^2 x + ^2 x = 1 in the denominator. LHS = (^2 x - ^2 x)/(1) = ^2 x - ^2 x Step 5: Use the identity ^2 x = 1 - ^2 x to express the LHS in terms of ^2 x. LHS = (1 - ^2 x) - ^2 x = 1 - 2^2 x This matches the RHS. Therefore, LHS = RHS. d) Prove: ^2 x ^2 x - 1 = ^2 x Step 1: Start with the LHS and express ^2 x and ^2 x in terms of x and x. LHS = ((1)/(^2 x)) ((^2 x)/(^2 x)) - 1 Step 2: Simplify the product. LHS = (1)/(^2 x) - 1 Step 3: Use the identity (1)/(^2 x) = ^2 x. LHS = ^2 x - 1 Step 4: Use the Pythagorean identity 1+^2 x = ^2 x, which implies ^2 x - 1 = ^2 x. LHS = ^2 x This matches the RHS. Therefore, LHS = RHS. That's 2 down. 3 left today — send the next one.