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If Xy + Yz + Zx = 1 Then Tan 1(x + Tan 1 Y + Tan 1 Z Is Equal To)

If Xy + Yz + Zx = 1 Then Tan 1(x + Tan 1 Y + Tan 1 Z Is Equal To)

You’ve been asked to help another student out with his math homework, and he sent you the following question: If Xy + Yz + Zx = 1, then tan 1(x + tan 1 Y + tan 1 Z) is equal to...? 

Now that you’ve figured out what the question says, how do you go about solving it? That’s what this article will answer.

Step 1: Simplify

Xy + Yz + Zx = 1, Step 2: x+tan1y+tan1z=0, Step 3: x-tan1y=0, Step 4: tan1x-tan1y=0, Step 5: -cotan2z=-cosec2Z, Step 6: -cotan2Z=-sin2Z. Dividing both sides by sin 2Z we get cotan 2Z/sin 2Z=cosec2N.

Step 2: Linear combinations

We are presented with three lines that intersect at a single point, thus we can assign an equation to each line and combine these equations by subtracting. 
These resulting equations are called linear combinations. The trick is figuring out which equation to assign to which line. 
In order for it make sense, we must consider what happens when you add or subtract a multiple of one equation from another one.

For example: 

So, if you added 2xy - yz + zx all together (or subtracted), you would end up with xy - yz - zx. 
Thus, it makes sense that xy will be assigned to Line 2, yz will be assigned to Line 3 and zx will be assigned to Line 4 in that exact order!

Step 3: Identify the variables

x, y, z, and tan1 (included in brackets above). Step 4: Find an equation to eliminate one of these variables. 
In other words, we need an equation that will give us a single-variable expression for only one of these variables. 
Let's try tan1, because it doesn't have any variables attached to it: tan1 x + tan1 y + tan1 z = 0 Step 5: Plug all of our known values into that equation (e.g., use x=0), and solve for that variable.

Step 4: Set up the equation

Determine what if-then statement you're solving for (in our case, it's z). 
In math terms, an if-then statement is written as if condition then outcome, and when we take that formula and plug it into an equation, we get a linear equation. 
So we'll set up our linear equation using those two elements in one of these two ways: x=Z or x+Z=1. 
Both will give us a simple linear equation to solve. The only difference is how they are interpreted.

Step 5: Solutions set 

Solving tan(t) equals 3, we have t equals 2. Step 6: Solving tan (2) is equal to 1, we have tan 2 equals √3/2. Therefore x=1, y=1 and z=√3/2.


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