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


Consider the following triangle:

enter image description here

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


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Updated on August 20, 2022


  • Juan
    Juan about 1 year

    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!

    • lulu
      lulu over 5 years
      Hint: draw a right triangle with legs $1,x$.
    • kingW3
      kingW3 over 5 years
      $\tan(\arctan(x))=\frac{\sin(\arctan x)} {\cos(\arctan x)} =x$ and $\sin^2\arctan x+\cos^2\arctan x=1$
    • Narasimham
      Narasimham over 5 years
      @Juan Draw a Pythagorean triangle with x opposit to $ \alpha=$ arctan $x$ and $1$ adjacent side, hypotenuse being $\sqrt{1+x^2}$ Can you now take it from here?