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
Step 1: Express 15^ as a difference of standard angles. We can write 15^ as 45^ - 30^. We will use the tangent subtraction formula. Step 2: Apply the tangent subtraction formula. The formula for (A - B) is: (A - B) = ( A - B)/(1 + A B) Substitute A = 45^ and B = 30^: (15^) = (45^ - 30^) = ((45^) - (30^))/(1 + (45^) (30^)) Step 3: Substitute the known exact values for (45^) and (30^). We know that (45^) = 1 and (30^) = (1)/(sqrt(3)). Substitute these values into the expression: (15^) = (1 - 1)/(sqrt(3))1 + (1) ((1)/(sqrt(3))) (15^) = (1 - 1)/(sqrt(3))1 + (1)/(sqrt(3)) Step 4: Simplify the expression by rationalizing the denominator. To simplify, multiply the numerator and the denominator by sqrt(3): (15^) = sqrt(3)(1 - (1)/(sqrt(3)))sqrt(3)(1 + (1)/(sqrt(3))) (15^) = sqrt(3) - 1sqrt(3) + 1 Now, rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (sqrt(3) - 1): (15^) = (sqrt(3) - 1)(sqrt(3) - 1)(sqrt(3) + 1)(sqrt(3) - 1) Expand the numerator and denominator: Numerator: (sqrt(3) - 1)^2 = (sqrt(3))^2 - 2(sqrt(3))(1) + (1)^2 = 3 - 2sqrt(3) + 1 = 4 - 2sqrt(3) Denominator: (sqrt(3) + 1)(sqrt(3) - 1) = (sqrt(3))^2 - (1)^2 = 3 - 1 = 2 Substitute these back into the expression: (15^) = 4 - 2sqrt(3)2 Factor out 2 from the numerator: (15^) = 2(2 - sqrt(3))2 Cancel out the 2 in the numerator and denominator: (15^) = 2 - sqrt(3) Thus, we have shown that (15^) = 2 - sqrt(3). (15^) = 2 - sqrt(3) That's 2 down. 3 left today — send the next one.