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
Here are the proofs for the trigonometric identities: i) Prove ( x)/(1+ x) + (1+ x)/( x) = 2 x Step 1: Combine the fractions on the left-hand side (LHS) using a common denominator, which is (1+ x)( x). LHS = ( x · x)/((1+ x) x) + ((1+ x)(1+ x))/((1+ x) x) LHS = (^2 x + (1+ x)^2)/((1+ x) x) Step 2: Expand the term (1+ x)^2 in the numerator. (1+ x)^2 = 1^2 + 2(1)( x) + ( x)^2 = 1 + 2 x + ^2 x Substitute this back into the numerator: LHS = (^2 x + 1 + 2 x + ^2 x)/((1+ x) x) Step 3: Apply the Pythagorean identity ^2 x + ^2 x = 1 to simplify the numerator. LHS = ((^2 x + ^2 x) + 1 + 2 x)/((1+ x) x) LHS = (1 + 1 + 2 x)/((1+ x) x) LHS = (2 + 2 x)/((1+ x) x) Step 4: Factor out 2 from the numerator. LHS = (2(1+ x))/((1+ x) x) Step 5: Cancel the common term (1+ x) from the numerator and denominator. LHS = (2)/( x) Step 6: Use the reciprocal identity x = (1)/( x). LHS = 2 x This matches the right-hand side (RHS). ( x)/(1+ x) + (1+ x)/( x) = 2 x ii) Prove ( x + x)/(1+ x) = x Step 1: Rewrite x in terms of x and x on the left-hand side (LHS). LHS = ( x + x)/( x)1+ x Step 2: Combine the terms in the numerator by finding a common denominator for the numerator. LHS = ( x x)/( x) + ( x)/( x)1+ x LHS = ( x x + x)/( x)1+ x Step 3: Factor out x from the numerator's numerator. LHS = ( x ( x + 1))/( x)1+ x Step 4: Rewrite the complex fraction as a multiplication. LHS = ( x ( x + 1))/( x) · (1)/(1+ x) Step 5: Cancel the common term ( x + 1) from the numerator and denominator. LHS = ( x)/( x) Step 6: Use the identity x = ( x)/( x). LHS = x This matches the right-hand side (RHS). ( x + x)/(1+ x) = x 3 done, 2 left today. You're making progress.