Value of (cos 135° · sin 210° · cos 150°) / (sin 270° · cos 315°) is

For angles greater than 90°, any trigonometric ratio can be evaluated by expressing the angle in terms of a reference angle from {30°, 45°, 60°, 90°} and applying the quadrant sign rule (ASTC): sine is positive in the second quadrant, tangent in the third, and cosine in the fourth, with all ratios positive in the first quadrant; every other combination in that quadrant is negative.

1. cos 135° = cos(180° − 45°) = −cos 45° = −1/√2, since 135° lies in the second quadrant where cosine is negative.

2. sin 210° = sin(180° + 30°) = −sin 30° = −1/2, since 210° lies in the third quadrant where sine is negative.

3. cos 150° = cos(180° − 30°) = −cos 30° = −√3/2, since 150° lies in the second quadrant where cosine is negative.

4. sin 270° = −1 and cos 315° = cos(360° − 45°) = cos 45° = 1/√2, since 315° lies in the fourth quadrant where cosine is positive.

5. Numerator = (−1/√2) · (−1/2) · (−√3/2) = −√3/(4√2), since two of the three negative signs cancel and one remains.

6. Denominator = (−1) · (1/√2) = −1/√2.

7. Dividing, the two negative signs cancel: [−√3/(4√2)] ÷ [−1/√2] = [√3/(4√2)] × √2 = √3/4.

As a check, using decimal approximations: cos 135° ≈ −0.7071, sin 210° = −0.5, cos 150° ≈ −0.8660 gives a numerator of about −0.3062; sin 270° = −1 and cos 315° ≈ 0.7071 give a denominator of about −0.7071; the ratio is about 0.4330, which matches √3/4 ≈ 0.4330.