# Simpliyfing an Expression of arctan(x) + arctan(y) + arctan(z)

I have an expression such as $$1.5arctan(x) - arctan(y) - arctan(1) = 0$$

Is there a way to simplify this equation by using the fact that

$$arctan(x) + arctan(y) + arctan(z) = arctan(\frac{x+y+z−xyz}{1−xy−yz−zx})$$

https://www.math-only-math.com/arctan-x-plus-arctan-y-plus-arctan-z.html

Edit: Its possible to use any other method to simplify the above expression. I just wanted to give an example. The site has more identities

• The sum of arctangents identity is true up to an integer multiple of $$\pi$$ (unless there's more information about $$x,y$$ and $$z$$). For example, when $$x=y=z=1$$, the two sides differ by $$\pi$$.
• You also need to have $$xy+yz+zx\ne 1$$ for the arctangent to be defined. So you need to check whether that can happen.
• You also have $$\arctan(1)=\frac{\pi}{4}$$ (unless the $$1$$ is a typo?), so you can move that on the other side.
• As for the simplification, rewrite $$1.5\arctan(x)$$ as $$\frac12\arctan(x) + \arctan(x)$$. Then use $$\arctan(x)=2\arctan\frac{x}{1+\sqrt{1+x^2}}$$ for $$\frac12\arctan(x)$$, and note that $$-\arctan(y)=\arctan(-y)$$. You'll then get the sum of $$3$$ arctangents, so you can simplify using that identity.