To solve this, we will calculate each trigonometric function individually:
cos(135°) = cos(180° - 45°) = -cos(45°) = -√2/2
sin(8π/3) = sin(4π + 4π/3) = sin(4π/3) = -√3/2
tan(7π/3) = tan(6π/3 + π/3) = tan(π/3) = √3
cos^2(π/8) = (cos(π/8))^2 = (sqrt(2+sqrt(2))/2)^2 = (2+sqrt(2))/4
sin^2(π/8) = (sin(π/8))^2 = (sqrt(2-sqrt(2))/2)^2 = (2-sqrt(2))/4
Now, we substitute these values into the expression:-√2/2 -√3/2 √3 (2+√2)/4 - (2-√2)/4 = = (3√2/4) (2 + √2) - (2 - √2)/4 == (3√2/2) + (3√2) - (2 - √2)/4 == 3√2/2 + 3√2 - 2 + √2 / 4= (5√2 + √2 ) / 2 - 2= 6√2 / 2 - 2= 3√2 - 2
Therefore, cos 135 sin 8П/3 tg 7П/3 cos^2 П/ 8 - sin^2 П/8 equals 3√2 - 2.
To solve this, we will calculate each trigonometric function individually:
cos(135°) = cos(180° - 45°) = -cos(45°) = -√2/2
sin(8π/3) = sin(4π + 4π/3) = sin(4π/3) = -√3/2
tan(7π/3) = tan(6π/3 + π/3) = tan(π/3) = √3
cos^2(π/8) = (cos(π/8))^2 = (sqrt(2+sqrt(2))/2)^2 = (2+sqrt(2))/4
sin^2(π/8) = (sin(π/8))^2 = (sqrt(2-sqrt(2))/2)^2 = (2-sqrt(2))/4
Now, we substitute these values into the expression:
-√2/2 -√3/2 √3 (2+√2)/4 - (2-√2)/4 =
= (3√2/4) (2 + √2) - (2 - √2)/4 =
= (3√2/2) + (3√2) - (2 - √2)/4 =
= 3√2/2 + 3√2 - 2 + √2 / 4
= (5√2 + √2 ) / 2 - 2
= 6√2 / 2 - 2
= 3√2 - 2
Therefore, cos 135 sin 8П/3 tg 7П/3 cos^2 П/ 8 - sin^2 П/8 equals 3√2 - 2.