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!

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

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calculustrigonometry

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!

## Best Answer

Consider the following triangle:

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