(d) Calculate each of the following without the use of a calculator: (1) \( \cos 80^{\circ} \cos 10^{\circ}-\sin 80^{\circ} \sin 10^{\circ} \) (2) \( \cos 70^{\circ} \cos 40^{\circ}+\sin 70^{\circ} \sin 40^{\circ} \) (3) \( \sin 280^{\circ} \cos 160^{\circ}-\cos 100^{\circ} \sin 200^{\circ} \) (4) \( \cos 265^{\circ} \sin 355^{\circ}-\sin 85^{\circ} \cos 175^{\circ} \) (5) \( \cos 65^{\circ} \cos 295^{\circ}-\sin 115^{\circ} \cos 205^{\circ} \) (6) \( \cos 50^{\circ} \sin 260^{\circ}+\cos 10^{\circ} \sin 140^{\circ} \)

To solve each of the trigonometric expressions, we will use the appropriate trigonometric identities. ### (1) \( \cos 80^{\circ} \cos 10^{\circ} - \sin 80^{\circ} \sin 10^{\circ} \) This expression can be simplified using the cosine of a sum identity: \[ \cos(A) \cos(B) - \sin(A) \sin(B) = \cos(A + B) \] Here, \( A = 80^{\circ} \) and \( B = 10^{\circ} \): \[ \cos(80^{\circ}) \cos(10^{\circ}) - \sin(80^{\circ}) \sin(10^{\circ}) = \cos(80^{\circ} + 10^{\circ}) = \cos(90^{\circ}) = 0 \] ### (2) \( \cos 70^{\circ} \cos 40^{\circ} + \sin 70^{\circ} \sin 40^{\circ} \) This expression can be simplified using the cosine of a difference identity: \[ \cos(A) \cos(B) + \sin(A) \sin(B) = \cos(A - B) \] Here, \( A = 70^{\circ} \) and \( B = 40^{\circ} \): \[ \cos(70^{\circ}) \cos(40^{\circ}) + \sin(70^{\circ}) \sin(40^{\circ}) = \cos(70^{\circ} - 40^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \] ### (3) \( \sin 280^{\circ} \cos 160^{\circ} - \cos 100^{\circ} \sin 200^{\circ} \) This expression can be simplified using the sine of a difference identity: \[ \sin(A) \cos(B) - \cos(A) \sin(B) = \sin(A - B) \] Here, \( A = 280^{\circ} \) and \( B = 160^{\circ} \): \[ \sin(280^{\circ}) \cos(160^{\circ}) - \cos(100^{\circ}) \sin(200^{\circ}) = \sin(280^{\circ} - 160^{\circ}) = \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \] ### (4) \( \cos 265^{\circ} \sin 355^{\circ} - \sin 85^{\circ} \cos 175^{\circ} \) Using the sine of a difference identity: \[ \sin(A) \cos(B) - \cos(A) \sin(B) = \sin(A - B) \] Here, \( A = 355^{\circ} \) and \( B = 265^{\circ} \): \[ \cos(265^{\circ}) \sin(355^{\circ}) - \sin(85^{\circ}) \cos(175^{\circ}) = \sin(355^{\circ} - 265^{\circ}) = \sin(90^{\circ}) = 1 \] ### (5) \( \cos 65^{\circ} \cos 295^{\circ} - \sin 115^{\circ} \cos 205^{\circ} \) Using the cosine of a sum identity: \[ \cos(A) \cos(B) - \sin(A) \sin(B) = \cos(A + B) \] Here, \( A = 65^{\circ} \) and \( B = 295^{\circ} \): \[ \cos(65^{\circ}) \cos(295^{\circ}) - \sin(115^{\circ}) \cos(205^{\circ}) = \cos(65^{\circ} + 295^{\circ}) = \cos(360^{\circ}) = 1 \] ### (6) \( \cos 50^{\circ} \sin 260^{\circ} + \cos 10^{\circ} \sin 140^{\circ} \) Using the sine of a sum identity: \[ \sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B) \] Here, \( A = 260^{\circ} \) and \( B = 50^{\circ} \): \[ \cos(50^{\circ}) \sin(260^{\circ}) + \cos(10^{\circ}) \sin(140^{\circ}) = \sin(260^{\circ} + 50^{\circ}) = \sin(310^{\circ}) = -\frac{\sqrt{3}}{2} \] ### Summary of Results: 1. \( 0 \) 2. \( \frac{\sqrt{3}}{2} \) 3. \( \frac{\sqrt{3}}{2} \) 4. \( 1 \) 5. \( 1 \) 6. \( -\frac{\sqrt{3}}{2} \)