Question

Cos 22,5° cos 22,5 ° - sin 22,5° sin 22,5 ° = .... sin 182 ° sin 28 ° - cos 178° cos 28 ° = .....

Answer 1

1)

$cos(22,5^{\circ})*cos(22,5^{\circ})-sin(22,4^{\circ})*sin(22,5^{\circ}) \\ \\ cos(22,5^{\circ})^{2}-sin(22,5^{\circ})^{2}=cos(45^{\circ})= \frac{ \sqrt{2} }{2}$

2)

$sin(182^{\circ})*sin(28^{\circ})-cos(178^{\circ})*cos(28^{\circ}) \\ \\ \frac{1}{2}*(cos(154^{\circ})-cos(210^{\circ}))- \frac{1}{2}*(cos(150^{\circ})+cos(206^{\circ}))= \\ \\ = \frac{1}{2}*(cos(154^{\circ})-(- \frac{ \sqrt{3} }{2}))- \frac{1}{2} *(- \frac{ \sqrt{3} }{2} +cos(206^{\circ}))= \\ \\ = \frac{cos(154^{\circ})}{2}+ \frac{ \sqrt{3} }{4} + \frac{ \sqrt{3} }{4}- \frac{cos(206^{\circ})}{2} = \frac{cos(154^{\circ})+ \sqrt{3}-cos(206^{\circ}) }{2}$

Теперь за формулой разницы косинусов:

$\frac{-2*sin(180^{\circ})*sin(26^{\circ})+ \sqrt{3} }{2} = \frac{-2*0*sin(-26^{\circ})+ \sqrt{3} }{2} = \frac{ \sqrt{3} }{2}$

Answer 2

To simplify the first expression, we can use the trigonometric identity:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Applying this identity with a = b = 22.5°, we get:

cos 22.5° cos 22.5° - sin 22.5° sin 22.5° = cos(22.5° - 22.5°) = cos 0° = 1

Therefore, the first expression simplifies to 1.

For the second expression, we can again use the trigonometric identity:

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Applying this identity with a = 182° and b = 28°, we get:

sin 182° sin 28° - cos 178° cos 28° = sin(182° - 28°) = sin 154°

Therefore, the second expression simplifies to sin 154°.