Okay, let's see. I need to find the value of x between 0 and 180 degrees such that tan 53° times tan 81° times tan x° equals the sum of tan 53°, tan 81°, and tan x°. Hmm, trigonometric equation involving tangents. Alright, let's start by writing down the equation:

tan 53° * tan 81° * tan x° = tan 53° + tan 81° + tan x°.

I need to solve for x. First, maybe I can simplify the equation by dividing both sides by tan x°, assuming tan x° isn't zero, but since x is between 0 and 180, tan x° is zero only at 0° and 180°, which are excluded. So dividing both sides by tan x° is safe.

That gives:

tan 53° * tan 81° = (tan 53° + tan 81° + tan x°) / tan x°.

Hmm, not sure if that helps. Let me think. Alternatively, maybe let's denote A = 53°, B = 81°, and C = x° for simplicity. Then the equation becomes:

tan A * tan B * tan C = tan A + tan B + tan C.

We can rearrange this equation:

tan A * tan B * tan C - tan C = tan A + tan B.

Factor out tan C on the left:

tan C (tan A * tan B - 1) = tan A + tan B.

Then, solve for tan C:

tan C = (tan A + tan B) / (tan A * tan B - 1).

So, substituting back A = 53° and B = 81°:

tan x° = [tan 53° + tan 81°] / [tan 53° * tan 81° - 1].

Now, maybe compute the numerator and denominator separately.

First, let's compute tan 53° and tan 81°. I remember that tan 45° is 1, tan 60° is about 1.732, so 53° is a bit more than 45°, so its tangent should be around 1.333? Wait, no, wait. Wait, 53° is in the first quadrant, so tan is positive. Let me recall exact values or approximate.

Alternatively, maybe use a calculator? But since this is a problem-solving scenario, maybe exact values can be used. Wait, 53° and 81°, those aren't standard angles, but maybe there's a relationship between them.

Wait, 53° and 81°... 53 + 81 = 134°, which isn't particularly special. Hmm. Alternatively, perhaps consider the tangent addition formula. Wait, if we have tan(A + B) = (tan A + tan B)/(1 - tan A tan B). Hmm, so if tan(A + B) = [tan A + tan B]/[1 - tan A tan B], then [tan A + tan B] = tan(A + B)(1 - tan A tan B).

Comparing that to our expression for tan x°, which is [tan A + tan B]/[tan A tan B - 1]. Let's see:

tan x° = [tan A + tan B]/(tan A tan B - 1) = - [tan A + tan B]/(1 - tan A tan B) = - tan(A + B).

Because tan(A + B) = [tan A + tan B]/[1 - tan A tan B], so [tan A + tan B]/[1 - tan A tan B] = tan(A + B), so [tan A + tan B]/[tan A tan B - 1] = -tan(A + B).

Therefore, tan x° = -tan(A + B). But tan is periodic with period 180°, so tan x° = tan(- (A + B) + 180°k) for some integer k. Since x is between 0 and 180°, let's compute A + B = 53° + 81° = 134°. Therefore, tan x° = -tan 134°. But tan 134° is equal to tan(180° - 46°) = -tan 46°. So tan x° = -(-tan 46°) = tan 46°. Therefore, x° is 46°.

Wait, let me check that again. If tan x° = -tan 134°, then tan x° = tan(-134° + 180°) = tan 46°. So x = 46°. That seems straightforward. Let me verify.

Alternatively, maybe I made a miscalculation. Let me go through the steps again.

We have:

tan x° = [tan 53° + tan 81°]/(tan 53° tan 81° - 1).

From the tangent addition formula:

tan(A + B) = (tan A + tan B)/(1 - tan A tan B).

Therefore, tan(A + B) = [tan A + tan B]/(1 - tan A tan B).

So rearranged, [tan A + tan B] = tan(A + B)(1 - tan A tan B).

Therefore, [tan A + tan B]/(tan A tan B - 1) = [tan(A + B)(1 - tan A tan B)]/(tan A tan B - 1) = -tan(A + B).

Hence, tan x° = -tan(A + B) = -tan(53° + 81°) = -tan(134°). Now, tan(134°) is equal to tan(180° - 46°) = -tan(46°), so tan(134°) = -tan(46°). Therefore, -tan(134°) = tan(46°). Therefore, tan x° = tan(46°), so x = 46°, since 46° is between 0 and 180°, and tan is injective (one-to-one) in that interval except at 90°, where it's undefined. Since 46° is in the first quadrant, x = 46° is the solution.

Let me check this by plugging back into the original equation.

Compute tan 53°, tan 81°, tan 46°, and then compute the left and right sides.

First, approximate the values:

tan 53° ≈ 1.3270 (since tan 45° = 1, tan 60° ≈ 1.732, 53° is closer to 45°, so around 1.327)

tan 81° ≈ 6.3138 (since tan 75° ≈ 3.732, tan 80° ≈ 5.671, tan 81° ≈ 6.3138)

tan 46° ≈ 1.0355 (since tan 45° = 1, tan 46° is slightly more, around 1.0355)

Left side: tan 53° * tan 81° * tan 46° ≈ 1.3270 * 6.3138 * 1.0355 ≈ Let's compute step by step.

First, 1.3270 * 6.3138 ≈ 1.3270 * 6 = 7.962, 1.3270 * 0.3138 ≈ approx 0.416. So total ≈ 7.962 + 0.416 ≈ 8.378.

Then multiply by 1.0355: 8.378 * 1.0355 ≈ 8.378 + 8.378*0.0355 ≈ 8.378 + 0.296 ≈ 8.674.

Right side: tan 53° + tan 81° + tan 46° ≈ 1.3270 + 6.3138 + 1.0355 ≈ 1.327 + 6.3138 = 7.6408 + 1.0355 ≈ 8.6763.

The left side is approximately 8.674 and the right side is approximately 8.6763, which are very close, considering the approximations. So this checks out. Therefore, x = 46° is the correct solution.

Alternatively, maybe there's another approach. Let's see. Suppose we use the identity that if tan A tan B tan C = tan A + tan B + tan C, then A + B + C = 180°, but wait, that's not necessarily always the case. Wait, actually, the identity is tan A + tan B + tan C = tan A tan B tan C if A + B + C = 180°, but in this case, the equation is tan A tan B tan C = tan A + tan B + tan C, which is the same as the identity if A + B + C = 180°, but in our problem, A is 53°, B is 81°, and C is x°. So if 53 + 81 + x = 180°, then x = 46°, which is exactly the solution we found. Therefore, this is another way to see it: if the sum of the angles is 180°, then tan A tan B tan C = tan A + tan B + tan C. Therefore, x must be 46°. That's a neat identity. So the answer is 46°.

But just to make sure, let's think if there could be another solution. Suppose, for example, that A + B + C = 180° + 180°k, but since all angles are between 0° and 180°, the only possible k is 0 or -1. If k = 0, then A + B + C = 180°, which gives x = 46°. If k = -1, then A + B + C = -180°, which is impossible because all angles are positive. Therefore, the only solution is x = 46°. So that's the answer.

Therefore, after verifying through multiple approaches—using the tangent addition formula, solving algebraically, and checking the trigonometric identity—it's clear that x is 46 degrees.

**Final Answer**
\boxed{46}
Therefore, the value of \( x \) that satisfies the equation is \(\boxed{46}\).
</think>