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 verifications for each identity: a) Verify (1 - x)/( x) = (1)/( x + x) Step 1: Start with the left-hand side (LHS) and rewrite it using fundamental identities. LHS = (1 - x)/( x) = (1)/( x) - ( x)/( x) Step 2: Express in terms of x and x. LHS = x - x Step 3: Multiply the expression by ( x + x)/( x + x). LHS = ( x - x) (( x + x)/( x + x)) Step 4: Use the difference of squares formula (a-b)(a+b) = a^2 - b^2 in the numerator. LHS = (^2 x - ^2 x)/( x + x) Step 5: Apply the Pythagorean identity ^2 x - ^2 x = 1. LHS = (1)/( x + x) This matches the right-hand side (RHS). Therefore, the identity is verified. Verified b) Verify x x = 1 Step 1: Rewrite x in terms of x. x = (1)/( x) Step 2: Substitute this into the left-hand side (LHS). LHS = x ((1)/( x)) Step 3: Simplify the expression. LHS = 1 This matches the right-hand side (RHS). Therefore, the identity is verified. Verified c) Verify ( z)/( z) + ( z)/( z) = 1 Step 1: Rewrite z and z in terms of z and z. z = (1)/( z) z = (1)/( z) Step 2: Substitute these into the left-hand side (LHS). LHS = ( z)/(1) z + ( z)/(1) z Step 3: Simplify each term. LHS = z · z + z · z LHS = ^2 z + ^2 z Step 4: Apply the Pythagorean identity ^2 z + ^2 z = 1. LHS = 1 This matches the right-hand side (RHS). Therefore, the identity is verified. Verified d) Verify 2x + 2 ^2 x = 1 Step 1: Use the double angle identity for 2x that involves ^2 x. 2x = 1 - 2^2 x Step 2: Substitute this into the left-hand side (LHS). LHS = (1 - 2^2 x) + 2^2 x Step 3: Simplify the expression. LHS = 1 - 2^2 x + 2^2 x LHS = 1 This matches the right-hand side (RHS). Therefore, the identity is verified. Verified e) Verify (1)/(2)x = ( x)/(1 + x) Step 1: Start with the right-hand side (RHS) and express x and x in terms of x and x. RHS = ( x)/(1 + x) = ( x)/( x)1 + (1)/( x) Step 2: Simplify the denominator by finding a common denominator. RHS = ( x)/( x)( x)/( x) + (1)/( x) = ( x)/( x)( x + 1)/( x) Step 3: Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator. RHS = ( x)/( x) · ( x)/( x + 1) Step 4: Cancel out x. RHS = ( x)/(1 + x) Step 5: This expression is a known half-angle identity for (x)/(2). (1)/(2)x = ( x)/(1 + x) Since the RHS simplifies to ( x)/(1 + x), which is equal to (1)/(2)x, the identity is verified. Verified