26. प्रमाणित गनुहोस् । (Prove that): \( \sin (\mathrm{A}+\mathrm{B})-\sin (\mathrm{A}-\mathrm{B})=2 \cos \mathrm{~A} \cdot \sin \mathrm{~B} \) 27. एडटा विस्थापनले बिन्दु \( (-3,-4) \) लाई \( (-7,0) \) मा विस्थापित गई, । बिन्दु \( (0,6) \) को आकति त्यही विस्थापन अनुसार पत्ता लगाउनुहोस् । ( A translation T transiates a point \( (-3,-4) \) to the point \( (-7,0) \). Find the image of the point \( (0,6) \) under this translation.)

Verify the identity by following steps: - step0: Verify: \(\sin\left(A+B\right)-\sin\left(A-B\right)=2\cos\left(A\right)\sin\left(B\right)\) - step1: Choose a side to work on: \(2\cos\left(A\right)\sin\left(B\right)=2\cos\left(A\right)\sin\left(B\right)\) - step2: Verify the identity: \(\textrm{true}\) Let's solve the two problems step by step. ### Problem 26: Prove that \[ \sin (A+B) - \sin (A-B) = 2 \cos A \cdot \sin B \] **Step 1: Use the sine addition and subtraction formulas.** The sine addition formula states: \[ \sin (A + B) = \sin A \cos B + \cos A \sin B \] The sine subtraction formula states: \[ \sin (A - B) = \sin A \cos B - \cos A \sin B \] **Step 2: Substitute these formulas into the left-hand side of the equation.** \[ \sin (A + B) - \sin (A - B) = (\sin A \cos B + \cos A \sin B) - (\sin A \cos B - \cos A \sin B) \] **Step 3: Simplify the expression.** \[ = \sin A \cos B + \cos A \sin B - \sin A \cos B + \cos A \sin B \] \[ = 2 \cos A \sin B \] **Conclusion:** Thus, we have shown that: \[ \sin (A + B) - \sin (A - B) = 2 \cos A \cdot \sin B \] This proves the statement is true. --- ### Problem 27: Find the image of the point \( (0,6) \) under the translation that moves \( (-3,-4) \) to \( (-7,0) \). **Step 1: Determine the translation vector.** The translation vector \( T \) can be found by subtracting the coordinates of the original point from the coordinates of the new point: \[ T = (-7, 0) - (-3, -4) = (-7 + 3, 0 + 4) = (-4, 4) \] **Step 2: Apply the translation to the point \( (0, 6) \).** To find the image of the point \( (0, 6) \) under this translation, we add the translation vector \( T \) to the point: \[ (0, 6) + (-4, 4) = (0 - 4, 6 + 4) = (-4, 10) \] **Conclusion:** The image of the point \( (0, 6) \) under the translation is \( (-4, 10) \).