Okay, so I need to find the least positive angle θ such that tan θ equals this big fraction. The numerator and denominator are both combinations of cosines and sines of different angles. Let me write them down again to make sure I have them right.

Numerator: cos5° cos20° + cos35° cos50° - sin5° sin20° - sin35° sin50°

Denominator: sin5° cos20° - sin35° cos50° + cos5° sin20° - cos35° sin50°

Hmm, this looks complicated. Maybe I can simplify both the numerator and the denominator using trigonometric identities. Let me recall some identities. 

First, the cosine of a sum: cos(A + B) = cosA cosB - sinA sinB. Similarly, the sine of a sum: sin(A + B) = sinA cosB + cosA sinB. Also, cosine of difference: cos(A - B) = cosA cosB + sinA sinB. And sine of difference: sin(A - B) = sinA cosB - cosA sinB.

Looking at the numerator: cos5 cos20 + cos35 cos50 - sin5 sin20 - sin35 sin50. Let me group terms:

= (cos5 cos20 - sin5 sin20) + (cos35 cos50 - sin35 sin50)

Each of these groups looks like the cosine of a sum. Because cos(A + B) = cosA cosB - sinA sinB. So:

= cos(5 + 20) + cos(35 + 50)

= cos25° + cos85°

Wait, 35 + 50 is 85. So numerator simplifies to cos25° + cos85°.

Similarly, let's check the denominator: sin5 cos20 - sin35 cos50 + cos5 sin20 - cos35 sin50

Let me rearrange the terms:

= (sin5 cos20 + cos5 sin20) - (sin35 cos50 + cos35 sin50)

Each of these parentheses is the sine of a sum. Because sin(A + B) = sinA cosB + cosA sinB.

So first term: sin(5 + 20) = sin25°

Second term: sin(35 + 50) = sin85°

Therefore, denominator becomes sin25° - sin85°

So now, the original expression becomes:

tanθ = [cos25° + cos85°] / [sin25° - sin85°]

Hmm, okay. So now I have tanθ = [cos25 + cos85] / [sin25 - sin85]

I can try to simplify numerator and denominator further. Let's handle the numerator first: cos25 + cos85. There's a formula for the sum of cosines: cosA + cosB = 2 cos[(A+B)/2] cos[(A-B)/2]

Applying that here:

cos25 + cos85 = 2 cos[(25+85)/2] cos[(25-85)/2] = 2 cos55° cos(-30°)

But cos(-30°) is cos30°, which is √3/2. So:

= 2 * cos55° * (√3/2) = √3 cos55°

Similarly, the denominator: sin25 - sin85. There's a formula for the difference of sines: sinA - sinB = 2 cos[(A+B)/2] sin[(A-B)/2]

Applying that:

sin25 - sin85 = 2 cos[(25+85)/2] sin[(25-85)/2] = 2 cos55° sin(-30°)

Sin(-30°) is -1/2. So:

= 2 cos55° (-1/2) = -cos55°

Therefore, the expression becomes:

tanθ = [√3 cos55°] / [-cos55°] = -√3

So tanθ = -√3

But θ needs to be the least positive angle. The tangent is negative, so θ must be in a quadrant where tangent is negative, which is the second or fourth quadrant. Since we want the least positive angle, which is the smallest angle in the positive direction (counterclockwise) from the x-axis where tangent is -√3.

But tangent is periodic with period 180°, so tanθ = tan(θ + 180°n). The reference angle for tanθ = √3 is 60°, since tan60° = √3. Therefore, tanθ = -√3 corresponds to angles 180° - 60° = 120° and 180° + 60° = 240°, etc. The least positive angle is 120°, since 120° is in the second quadrant where tangent is negative, and it's the smallest such angle.

Wait, let me check: tan120° = tan(180° - 60°) = -tan60° = -√3. Yes, that's correct. So θ = 120°. Therefore, the least positive angle is 120 degrees.

But let me verify my steps again to make sure I didn't make a mistake.

Starting with numerator: cos25 + cos85. Using sum formula:

cosA + cosB = 2 cos[(A+B)/2] cos[(A-B)/2]

A=25, B=85, so (25+85)/2=55, (25-85)/2=-30. So 2 cos55 cos(-30) = 2 cos55 * (√3/2) = √3 cos55. Correct.

Denominator: sin25 - sin85. Using sinA - sinB = 2 cos[(A+B)/2] sin[(A-B)/2]

A=25, B=85, so (25+85)/2=55, (25-85)/2=-30. So 2 cos55 sin(-30) = 2 cos55 (-1/2) = -cos55. Correct.

Therefore, tanθ = (√3 cos55)/(-cos55) = -√3. Correct.

Therefore, θ is 180° - 60° = 120°, since tangent is negative there. Yes. So 120° is the answer.

But wait, let me check if there is a smaller positive angle. Since tangent is periodic with period 180°, but 120° is already the smallest positive angle where tanθ = -√3. Because if you subtract 180°, you get -60°, which is negative, so the least positive angle is indeed 120°.

Therefore, the answer should be 120°. Let me confirm once more.

Alternatively, maybe there's a different approach? Let me see.

Alternatively, maybe using complex numbers or vectors? Not sure. But the steps I took seem solid. Let me check if there's a different identity I could use.

Alternatively, maybe factor the numerator and denominator differently. Let's see.

Original numerator: cos5 cos20 + cos35 cos50 - sin5 sin20 - sin35 sin50

Group terms as (cos5 cos20 - sin5 sin20) + (cos35 cos50 - sin35 sin50) = cos(5+20) + cos(35+50) = cos25 + cos85. Correct.

Denominator: sin5 cos20 - sin35 cos50 + cos5 sin20 - cos35 sin50. Let's see, grouping as (sin5 cos20 + cos5 sin20) - (sin35 cos50 + cos35 sin50) = sin(5+20) - sin(35+50) = sin25 - sin85. Correct.

So yes, that's correct. So the expression simplifies to (√3 cos55)/(-cos55) = -√3. Therefore, θ = arctan(-√3). The arctangent of -√3 is -60°, but since we need the least positive angle, we add 180° to get 120°. So yes, 120° is correct.

I think that's thorough. Let me check with approximate values to see if it makes sense.

Compute numerator: cos5 cos20 + cos35 cos50 - sin5 sin20 - sin35 sin50

Using calculator:

cos5 ≈ 0.9962, cos20 ≈ 0.9397, cos35 ≈ 0.8192, cos50 ≈ 0.6428, sin5 ≈ 0.0872, sin20 ≈ 0.3420, sin35 ≈ 0.5736, sin50 ≈ 0.7660

Compute numerator:

0.9962*0.9397 ≈ 0.937

0.8192*0.6428 ≈ 0.527

0.0872*0.3420 ≈ 0.0298

0.5736*0.7660 ≈ 0.439

So numerator ≈ 0.937 + 0.527 - 0.0298 - 0.439 ≈ (0.937 + 0.527) - (0.0298 + 0.439) ≈ 1.464 - 0.4688 ≈ 0.9952

Denominator: sin5 cos20 - sin35 cos50 + cos5 sin20 - cos35 sin50

Compute each term:

sin5 cos20 ≈ 0.0872*0.9397 ≈ 0.0823

sin35 cos50 ≈ 0.5736*0.6428 ≈ 0.3683

cos5 sin20 ≈ 0.9962*0.3420 ≈ 0.3406

cos35 sin50 ≈ 0.8192*0.7660 ≈ 0.6293

So denominator ≈ 0.0823 - 0.3683 + 0.3406 - 0.6293

Compute step by step:

0.0823 - 0.3683 ≈ -0.286

-0.286 + 0.3406 ≈ 0.0546

0.0546 - 0.6293 ≈ -0.5747

So numerator ≈ 0.9952, denominator ≈ -0.5747

Therefore, tanθ ≈ 0.9952 / (-0.5747) ≈ -1.729

Compute arctangent of -1.729. Since tan60° ≈ 1.732, so arctan(-1.732) ≈ -60°, but since we need positive angle, add 180° to get 120°. So yes, the approximate calculation also gives 120°, which matches our previous result.

Therefore, I'm confident the answer is 120 degrees.

**Final Answer**
\boxed{120}
Therefore, the least positive angle $\theta$ satisfying the given equation is $\boxed{120^\circ}$.
</think>