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 solutions. 10. Find the exact value of (15^). Step 1: Express 15^ as a difference of standard angles. (15^) = (45^ - 30^) Step 2: Apply the tangent difference formula (A-B) = ((A)-(B))/(1+(A)(B)). = ((45^) - (30^))/(1 + (45^)(30^)) = (1 - 1)/(sqrt(3))1 + 1 · (1)/(sqrt(3)) = sqrt(3)-1sqrt(3)sqrt(3)+1sqrt(3) = sqrt(3)-1sqrt(3)+1 Step 3: Rationalize the denominator. = sqrt(3)-1sqrt(3)+1 · sqrt(3)-1sqrt(3)-1 = (sqrt(3))^2 - 2sqrt(3) + 1^2(sqrt(3))^2 - 1^2 = 3 - 2sqrt(3) + 13 - 1 = 4 - 2sqrt(3)2 = 2 - sqrt(3) The final answer is 2 - sqrt(3). 11. Prove that (3x) = 3(x) - 4^3(x). Start with the left-hand side (LHS): (3x) = (2x+x) Apply the sum formula for sine: (A+B) = (A)(B) + (A)(B). = (2x)(x) + (2x)(x) Apply double angle formulas: (2x) = 2(x)(x) and (2x) = 1-2^2(x). = (2(x)(x))(x) + (1-2^2(x))(x) = 2(x)^2(x) + (x) - 2^3(x) Substitute ^2(x) = 1-^2(x). = 2(x)(1-^2(x)) + (x) - 2^3(x) = 2(x) - 2^3(x) + (x) - 2^3(x) Combine like terms. = 3(x) - 4^3(x) This matches the right-hand side (RHS). The identity is proven.