(2) Hence show that \( \sin 43^{\circ}+\sin 17^{\circ}=\cos 13^{\circ} \)

To show that \( \sin 43^{\circ} + \sin 17^{\circ} = \cos 13^{\circ} \), we can use the sine addition formula and some trigonometric identities. ### Step 1: Use the sine addition formula The sine addition formula states that: \[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \] In our case, let \( A = 43^{\circ} \) and \( B = 17^{\circ} \). ### Step 2: Calculate \( A + B \) and \( A - B \) 1. Calculate \( A + B \): \[ A + B = 43^{\circ} + 17^{\circ} = 60^{\circ} \] 2. Calculate \( A - B \): \[ A - B = 43^{\circ} - 17^{\circ} = 26^{\circ} \] ### Step 3: Substitute into the sine addition formula Now substitute \( A + B \) and \( A - B \) into the sine addition formula: \[ \sin 43^{\circ} + \sin 17^{\circ = 2 \sin\left(\frac{60^{\circ}}{2}\right) \cos\left(\frac{26^{\circ}}{2}\right) \] This simplifies to: \[ \sin 43^{\circ} + \sin 17^{\circ} = 2 \sin(30^{\circ}) \cos(13^{\circ}) \] ### Step 4: Evaluate \( \sin(30^{\circ}) \) We know that: \[ \sin(30^{\circ}) = \frac{1}{2} \] ### Step 5: Substitute back into the equation Substituting this value back, we have: \[ \sin 43^{\circ} + \sin 17^{\circ} = 2 \cdot \frac{1}{2} \cos(13^{\circ}) = \cos(13^{\circ}) \] ### Conclusion Thus, we have shown that: \[ \sin 43^{\circ} + \sin 17^{\circ} = \cos 13^{\circ} \] This completes the proof.