e Determine, without using a calculator, the value of \( 1.1 \cos 150^{\circ} \cdot \cos \left(-28^{\circ}\right)+\cos 60^{\circ} \cdot \cos 62^{\circ} \) \( 1.2 \sin \left(85^{\circ}+\theta\right) \cdot \cos \left(25^{\circ}+\theta\right)-\cos \left(85^{\circ}+\theta\right) \cdot \sin \left(25^{\circ}+\theta\right) \) \( 1.3 \cos \left(A+55^{\circ}\right) \cos \left(A+10^{\circ}\right)+\sin \left(A+55^{\circ}\right) \sin \left(A+10^{\circ}\right) \) \( 1.4 \frac{\sin 21^{\circ} \cos w+\cos 21^{\circ} \sin w}{\sin \left(w+21^{\circ}\right)}-\tan ^{2} \beta \cdot \cos ^{2} \beta \).

Let's solve each part step by step. ### 1.1 \( 1.1 \cos 150^{\circ} \cdot \cos \left(-28^{\circ}\right)+\cos 60^{\circ} \cdot \cos 62^{\circ} \) **Step 1: Calculate \( \cos 150^{\circ} \)** Using the cosine of an angle in the second quadrant: \[ \cos 150^{\circ} = -\cos(180^{\circ} - 30^{\circ}) = -\cos 30^{\circ} = -\frac{\sqrt{3}}{2} \] **Step 2: Calculate \( \cos(-28^{\circ}) \)** Using the property of cosine: \[ \cos(-28^{\circ}) = \cos(28^{\circ} \] **Step 3: Calculate \( \cos 60^{\circ} \)** \[ \cos 60^{\circ} = \frac{1}{2} \] **Step 4: Calculate \( \cos 62^{\circ} \)** This value remains as is since we don't need to calculate it explicitly. **Step 5: Combine the results** Now substituting back into the expression: \[ 1.1 = -\frac{\sqrt{3}}{2} \cdot \cos(28^{\circ}) + \frac{1}{2} \cdot \cos(62^{\circ}) \] ### 1.2 \( 1.2 \sin \left(85^{\circ}+\theta\right) \cdot \cos \left(25^{\circ}+\theta\right)-\cos \left(85^{\circ}+\theta\right) \cdot \sin \left(25^{\circ}+\theta\right) \) **Step 1: Use the sine subtraction formula** This expression can be simplified using the sine subtraction formula: \[ \sin(A) \cos(B) - \cos(A) \sin(B) = \sin(A - B) \] Let \( A = 85^{\circ} + \theta \) and \( B = 25^{\circ} + \theta \): \[ 1.2 = \sin\left((85^{\circ} + \theta) - (25^{\circ} + \theta)\right) = \sin(60^{\circ}) \] **Step 2: Calculate \( \sin(60^{\circ}) \)** \[ \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \] ### 1.3 \( 1.3 \cos \left(A+55^{\circ}\right) \cos \left(A+10^{\circ}\right)+\sin \left(A+55^{\circ}\right) \sin \left(A+10^{\circ}\right) \) **Step 1: Use the cosine addition formula** This expression can be simplified using the cosine addition formula: \[ \cos(A) \cos(B) + \sin(A) \sin(B) = \cos(A - B) \] Let \( A = A + 55^{\circ} \) and \( B = A + 10^{\circ} \): \[ 1.3 = \cos\left((A + 55^{\circ}) - (A + 10^{\circ})\right) = \cos(45^{\circ}) \] **Step 2: Calculate \( \cos(45^{\circ}) \)** \[ \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \] ### 1.4 \( 1.4 \frac{\sin 21^{\circ} \cos w+\cos 21^{\circ} \sin w}{\sin \left(w+21^{\circ}\right)}-\tan ^{2} \beta \cdot \cos ^{2} \beta \) **Step 1: Use the sine addition formula** The first part can be simplified using the sine addition formula: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] Thus, \[ \frac{\sin(21^{\circ} + w)}{\sin(w + 21^{\circ})} = 1 \] **Step 2: The second part remains as is** So we have: \[ 1.4 = 1 - \tan^2 \beta \cdot \cos^2 \beta \] ### Final Results 1.1: \( -\frac{\sqrt{3}}{2} \cdot \cos(28^{\circ}) + \frac{1}{2} \cdot \cos(62^{\circ}) \) 1.2: \( \frac{\sqrt{3}}{2} \) 1.3: \( \frac{\sqrt{2}}{2} \) 1.4: \( 1 - \tan^2 \beta \cdot \cos^2 \beta \)