ocHyứre. 3. (cos^2(-125^0)sin(285^0))/(ctg^2)(324^(0)tg(165^0))

To solve the given expression, we need to simplify the trigonometric functions and then perform the division.<br /><br />Given expression: $\frac {cos^{2}(-125^{0})sin(285^{0})}{ctg^{2}(324^{0})tg(165^{0})}$<br /><br />Step 1: Simplify the trigonometric functions.<br />cos(-125°) = cos(360° - 125°) = cos(235°) = -cos(25°)<br />sin(285°) = sin(360° - 75°) = -sin(75°)<br />ctg(324°) = ctg(360° - 36°) = -ctg(36°)<br />tg(165°) = tg(180° - 15°) = -tg(15°)<br /><br />Step 2: Substitute the simplified trigonometric functions into the expression.<br />$\frac {(-cos(25°))^2 \cdot (-sin(75°))}{(-ctg(36°))^2 \cdot (-tg(15°))}$<br /><br />Step 3: Simplify the expression.<br />$\frac {cos^2(25°) \cdot sin(75°)}{ctg^2(36°) \cdot tg(15°)}$<br /><br />Step 4: Use the trigonometric identities to simplify the expression further.<br />cos^2(25°) = 1 - sin^2(25°)<br />sin(75°) = cos(15°)<br />ctg(36°) = $\frac{1}{tg(36°)}$<br />tg(15°) = $\frac{sin(15°)}{cos(15°)}$<br /><br />Substituting these identities into the expression, we get:<br />$\frac {(1 - sin^2(25°)) \cdot cos(15°)}{(\frac{1}{tg(36°)})^2 \cdot \frac{sin(15°)}{cos(15°)}}$<br /><br />Step 5: Simplify the expression further.<br />$\frac {(1 - sin^2(25°)) \cdot cos(15°)}{\frac{1}{tg^2(36°)} \cdot \frac{sin(15°)}{cos(15°)}}$<br /><br />Step 6: Multiply the numerator and denominator by $tg^2(36°) \cdot cos(15°)$.<br />$\frac {(1 - sin^2(25°)) \cdot cos(15°) \cdot tg^2(36°) \cdot cos(15°)}{1 \cdot sin(15°)}$<br /><br />Step 7: Simplify the expression.<br />$\frac {(1 - sin^2(25°)) \cdot cos^2(15°) \cdot tg^2(36°)}{sin(15°)}$<br /><br />Step 8: Use the trigonometric identities to simplify the expression further.<br />sin(15°) = $\frac{\sqrt{3} - 1}{2\sqrt{2}}$<br />cos^2(15°) = $\frac{1 + \sqrt{3}}{2\sqrt{2}}$<br />tg^2(36°) = $\frac{1 - \sqrt{3}}{2\sqrt{2}}$<br /><br />Substituting these identities into the expression, we get:<br />$\frac {(1 - sin^2(25°)) \cdot \frac{1 + \sqrt{3}}{2\sqrt{2}} \cdot \frac{1 - \sqrt{3}}{2\sqrt{2}}}{\frac{\sqrt{3} - 1}{2\sqrt{2}}}$<br /><br />Step 9: Simplify the expression further.<br />$\frac {(1 - sin^2(25°)) \cdot \frac{1 + \sqrt{3}}{2\sqrt{2}} \cdot \frac{1 - \sqrt{3}}{2\sqrt{2}}}{\frac{\sqrt{3} - 1}{2\sqrt{2}}}$<br /><br />Step 10: Multiply the numerator and denominator by $2\sqrt{2}$.<br />$\frac {(1 - sin^2(25°)) \cdot \frac{1 + \sqrt{3}}{2\sqrt{2}} \cdot \frac{1 - \sqrt{3}}{2\sqrt{2}} \cdot 2\sqrt{2}}{\frac{\sqrt{3} - 1}{2\sqrt{2}} \cdot 2\sqrt{2}}$<br /><br />Step 11: Simplify the expression.<br />$\frac {(1 - sin^2(25°)) \cdot \frac{1 + \sqrt{3}}{2} \cdot \frac{1 - \sqrt{3}}{2}}{\sqrt{3} - 1}$<br /><br />Step 12: Simplify the expression further.<br />$\frac {(1 - sin^2(25°