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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

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A couple of things:

  • 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.

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