Given constants $C$ and $D$, suppose that $ an A$ and $ an B$ are the solutions to $x^{2}+C x+D=0$, where $ an (A+B)$ is defined.

Simplify $\sin ^{2}(A+B)+C \sin (A+B) \cos (A+B)+D \cos ^{2}(A+B)$.

Question

Given constants $C$ and $D$, suppose that $	an A$ and $	an B$ are the solutions to $x^{2}+C x+D=0$, where $	an (A+B)$ is defined.

Simplify $\sin ^{2}(A+B)+C \sin (A+B) \cos (A+B)+D \cos ^{2}(A+B)$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Use the identity $\sin^2 x + \cos^2 x = 1$ to simplify the expression. ‖ ‖ step 2 ⋮ Substitute $\sin^2(A+B) + \cos^2(A+B)$ with 1 in the given expression. ‖ ‖ step 3 ⋮ The expression simplifies to $1 + C \sin(A+B) \cos(A+B) + D \cos^2(A+B)$. ‖ ‖ step 4 ⋮ Use the identity $\sin(2x) = 2 \sin x \cos x$ to simplify $C \sin(A+B) \cos(A+B)$. ‖ ‖ step 5 ⋮ Substitute $C \sin(A+B) \cos(A+B)$ with $\frac{C}{2} \sin(2(A+B))$ in the expression. ‖ ‖ step 6 ⋮ The expression now is $1 + \frac{C}{2} \sin(2(A+B)) + D \cos^2(A+B)$. ‖ ‖ step 7 ⋮ Use the quadratic equation $x^2 + Cx + D = 0$ to express $ an(A+B)$ in terms of $C$ and $D$. ‖ ‖ step 8 ⋮ Since $ an A$ and $ an B$ are roots, use Vieta's formulas: $ an A + an B = -\frac{C}{1}$ and $ an A \cdot an B = \frac{D}{1}$. ‖ ‖ step 9 ⋮ Use the identity $ an(A+B) = \frac{ an A + an B}{1 - an A an B}$ to find $ an(A+B)$. ‖ ‖ step 10 ⋮ Substitute $ an A + an B$ and $ an A \cdot an B$ into the identity to get $ an(A+B) = \frac{-C}{1-D}$. ‖ ‖ step 11 ⋮ Use the identity $ an^2 x + 1 = \sec^2 x$ to relate $ an(A+B)$ and $\cos^2(A+B)$. ‖ ‖ step 12 ⋮ Substitute $\cos^2(A+B)$ with $\frac{1}{\sec^2(A+B)}$ in the expression. ‖ ‖ step 13 ⋮ The expression now is $1 + \frac{C}{2} \sin(2(A+B)) + D \left(\frac{1}{\sec^2(A+B)} ight)$. ‖ ‖ step 14 ⋮ Substitute $\sec^2(A+B)$ with $1 + an^2(A+B)$ and then with $1 + \left(\frac{-C}{1-D} ight)^2$. ‖ ‖ step 15 ⋮ Simplify the expression to find the final simplified form. ‖ ∻Answer∻ ⚹ The final simplified form of the expression is $1 + \frac{C}{2} \sin(2(A+B)) + D \left(\frac{1}{1 + \left(\frac{-C}{1-D} ight)^2} ight)$. ⚹ ∻Key Concept∻ ⚹Trigonometric identities and Vieta's formulas⚹ ∻Explanation∻ ⚹The solution involves using trigonometric identities to simplify the given expression and applying Vieta's formulas to relate the sum and product of the roots of the quadratic equation to the coefficients.⚹◊What is the relationship between $\sin^2(A+B)$, $C\sin(A+B)\cos(A+B)$, and $D\cos^2(A+B)$?⍭ Generate me a similar question◊

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