Wrife ear hexpresilon as a single anglo arad evaluate if powable. 2.1 a sin \( 10^{\circ} \cos 30^{\circ} \) 2.) \( 1-2 \sin ^{21} 30^{+} \) \( 2.52 \cos 150^{\circ} \cdot \sin 300^{\circ} \) \( 2.2 \cos ^{3} 45^{\circ}-81 n^{2} 45 \) \( 242 \sin 22,5^{\circ}-\cos 22 \), \( 2.6 \quad 2 \cos ^{2} 15^{\circ}-1 \) \( 2 . \quad 4-8 y^{2}+15 \) \( 2.8 \quad 4 \sin 75^{\circ} \cdot \cos 75 \)

Let's evaluate each expression step by step, simplifying them where possible. ### 1. \( a = \sin(10^{\circ}) \cos(30^{\circ}) \) Using the known values: - \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) Thus, we can express \( a \) as: \[ a = \sin(10^{\circ}) \cdot \frac{\sqrt{3}}{2} \] ### 2. \( b = 1 - 2 \sin^{2}(30^{\circ}) \) Using the known value: - \( \sin(30^{\circ}) = \frac{1}{2} \) Thus, we can express \( b \) as: \[ b = 1 - 2 \left(\frac{1}{2}\right)^{2} = 1 - 2 \cdot \frac{1}{4} = 1 - \frac{1}{2} = \frac{1}{2} \] ### 3. \( c = 2.52 \cos(150^{\circ}) \cdot \sin(300^{\circ}) \) Using the known values: - \( \cos(150^{\circ}) = -\frac{\sqrt{3}}{2} \) - \( \sin(300^{\circ}) = -\frac{\sqrt{3}}{2} \) Thus, we can express \( c \) as: \[ c = 2.52 \cdot \left(-\frac{\sqrt{3}}{2}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right) = 2.52 \cdot \frac{3}{4} = \frac{7.56}{4} = 1.89 \] ### 4. \( d = \cos^{3}(45^{\circ}) - 81 n^{2} \) Using the known value: - \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \) Thus, we can express \( d \) as: \[ d = \left(\frac{\sqrt{2}}{2}\right)^{3} - 81 n^{2} = \frac{2\sqrt{2}}{8} - 81 n^{2} = \frac{\sqrt{2}}{4} - 81 n^{2} \] ### 5. \( e = 242 \sin(22.5^{\circ}) - \cos(22^{\circ}) \) Using the known value: - \( \sin(22.5^{\circ}) = \frac{\sqrt{2 - \sqrt{2}}}{2} \) Thus, we can express \( e \) as: \[ e = 242 \cdot \frac{\sqrt{2 - \sqrt{2}}}{2} - \cos(22^{\circ}) \] ### 6. \( f = 2 \cos^{2}(15^{\circ}) - 1 \) Using the known value: - \( \cos(15^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4} \) Thus, we can express \( f \) as: \[ f = 2 \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)^{2} - 1 \] ### 7. \( g = 4 - 8y^{2} + 15 \) This simplifies directly to: \[ g = 19 - 8y^{2} \] ### 8. \( h = 4 \sin(75^{\circ}) \cdot \cos(75^{\circ}) \) Using the known identity: \[ \sin(75^{\circ}) = \cos(15^{\circ}) \] Thus, we can express \( h \) as: \[ h = 4 \cdot \sin(75^{\circ}) \cdot \cos(75^{\circ}) = 2 \cdot \sin(150^{\circ}) = 2 \cdot \frac{1}{2} = 1 \] Now, let's evaluate the expressions that require numerical calculations. I will proceed to calculate the numerical values for \( a, c, e, f, \) and \( h \). Calculate the value by following steps: - step0: Calculate: \(2\left(\frac{\left(\sqrt{6}+\sqrt{2}\right)}{4}\right)^{2}-1\) - step1: Remove the parentheses: \(2\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^{2}-1\) - step2: Multiply the terms: \(\frac{2+\sqrt{3}}{2}-1\) - step3: Reduce fractions to a common denominator: \(\frac{2+\sqrt{3}}{2}-\frac{2}{2}\) - step4: Transform the expression: \(\frac{2+\sqrt{3}-2}{2}\) - step5: Remove the opposites: \(\frac{\sqrt{3}}{2}\) Calculate or simplify the expression \( \sin(10) * (\sqrt(3)/2) \). Calculate the value by following steps: - step0: Calculate: \(\sin\left(10\right)\left(\frac{\sqrt{3}}{2}\right)\) - step1: Remove the parentheses: \(\sin\left(10\right)\times \frac{\sqrt{3}}{2}\) - step2: Multiply: \(\frac{\sin\left(10\right)\times \sqrt{3}}{2}\) - step3: Multiply: \(\frac{\sqrt{3}\times \sin\left(10\right)}{2}\) - step4: Calculate the trigonometric value: \(-0.471136\) Calculate or simplify the expression \( 242 * (\sqrt(2 - \sqrt(2))/2) - \cos(22) \). Calculate the value by following steps: - step0: Calculate: \(242\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)-\cos\left(22\right)\) - step1: Remove the parentheses: \(242\times \frac{\sqrt{2-\sqrt{2}}}{2}-\cos\left(22\right)\) - step2: Reduce the numbers: \(121\sqrt{2-\sqrt{2}}-\cos\left(22\right)\) - step3: Calculate the trigonometric value: \(93.609351\) Here are the evaluated results for the expressions: ### 1. \( a = \sin(10^{\circ}) \cos(30^{\circ}) \) \[ a \approx -0.471136 \] ### 2. \( b = 1 - 2 \sin^{2}(30^{\circ}) \) \[ b = \frac{1}{2} \] ### 3. \( c = 2.52 \cos(150^{\circ}) \cdot \sin(300^{\circ}) \) \[ c \approx 1.89 \] ### 4. \( d = \cos^{3}(45^{\circ}) - 81 n^{2} \) \[ d = \frac{\sqrt{2}}{4} - 81 n^{2} \] ### 5. \( e = 242 \sin(22.5^{\circ}) - \cos(22^{\circ}) \) \[ e \approx 93.609351 \] ### 6. \( f = 2 \cos^{2}(15^{\circ}) - 1 \) \[ f = 2 \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)^{2} - 1 \] ### 7. \( g = 4 - 8y^{2} + 15 \) \[ g = 19 - 8y^{2} \] ### 8. \( h = 4 \sin(75^{\circ}) \cdot \cos(75^{\circ}) \) \[ h = 1 \] ### Summary of Results: - \( a \approx -0.471136 \) - \( b = \frac{1}{2} \) - \( c \approx 1.89 \) - \( d = \frac{\sqrt{2}}{4} - 81 n^{2} \) - \( e \approx 93.609351 \) - \( f = 2 \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)^{2} - 1 \) - \( g = 19 - 8y^{2} \) - \( h = 1 \) If you need further calculations or simplifications for any specific expression, please let me know!