Expressions for $\sin(\arctan(x))$ and $\cos(\arctan(x))$ that do not contain trigonometric functions

Question

Find expressions for $\sin(\arctan(x))$ and $\cos(\arctan(x))$ that do not contain trigonometric functions.

I have been trying to solve it for days, but I just can't figure it out!

Some help would be so nice!

Answer 1

Consider the following triangle:

Now: $$\tan\theta=\frac x1=x\implies \arctan x=\theta\\\sin\arctan x=\sin\theta=\frac{x}{\sqrt{1+x^2}}\\\cos\arctan x=\cos\theta=\frac{1}{\sqrt{1+x^2}}$$

Answer 2

Consider the right triangle with vertices $(0,0)$, $(1,0)$, and $(1,x)$. Then, if $\theta$ is the angle between the $x$-axis and the hypotenuse, you have: $$\tan \theta = x$$ So that: $$\arctan(x)=\theta$$ Now, you have: $$\sin(\arctan(x))=\frac{x}{\sqrt{1+x^2}}$$ and $$ \cos(\arctan(x))=\frac{1}{\sqrt{1+x^2}}$$