If A+ B=x and A+ B=y, then (A+B)=

Correct Answer is : 2xyx2+y2{2 x y}{x²+y²}x2+y22xy

Correct Answer is : 2xyx2+y2{2 x y}{x²+y²}x2+y22xy

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MHTCET 23 Apr 2025 Shift 1 Paper

Question : 111 of 150

Marks: +1, -0

\sin A+\sin B=x

\cos A+\cos B=y

\sin (A+B)=

Solution:

Concept:

Use sum-to-product identities and the tangent half-angle formula for sine.

Explanation:

Divide the two given equations:

\frac{\sin A+\sin B}{\cos A+\cos B}=\frac{x}{y}

Apply sum-to-product formulas:

\frac{2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)}{2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)}=\frac{x}{y}

Cancel the common factor

2\cos\left(\frac{A-B}{2}\right)

\tan\left(\frac{A+B}{2}\right)=\frac{x}{y}

Now use the identity:

\sin(A+B)=\frac{2\tan\left(\frac{A+B}{2}\right)}{1+\tan^2\left(\frac{A+B}{2}\right)}

Substitute

\tan\left(\frac{A+B}{2}\right)=\frac{x}{y}

\sin(A+B)=\frac{2\left(\frac{x}{y}\right)}{1+\left(\frac{x}{y}\right)^2}=\frac{2xy}{x^2+y^2}

Answer:

\frac{2xy}{x^2+y^2}

(Option A).

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