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. 3. Solve the equation ^2(x) + ^2(x) = 1 for all x. This is the Pythagorean trigonometric identity. The equation is true for all real values of x. The final answer is All real numbers. 4. Prove the identity (2x) = ^2(x) - ^2(x). Start with the sum formula for cosine: (A+B) = (A)(B) - (A)(B) Let A=x and B=x: (x+x) = (x)(x) - (x)(x) (2x) = ^2(x) - ^2(x) The identity is proven. 5. Prove that (A-B) = ((A) - (B)) / (1 + (A)(B)). Start with the definition of tangent and the difference formulas for sine and cosine: (A-B) = ((A-B))/((A-B)) = ((A)(B) - (A)(B))/((A)(B) + (A)(B)) Divide the numerator and denominator by (A)(B): = ((A)(B))/((A)(B)) - ((A)(B))/((A)(B))((A)(B))/((A)(B)) + ((A)(B))/((A)(B)) = ((A))/((A)) - ((B))/((B))1 + ((A))/((A))((B))/((B)) = ((A) - (B))/(1 + (A)(B)) The identity is proven.