Question

For questions 1 – 2, find an algebraic expression equivalent to the following expressions:

Answer 1

Let $y=\tan^{-1}x$, so that $\tan y=x$. Picture a right triangle with hypotenuse of length $z$ and a reference angle $y$. Then

$\cos y=\dfrac 1z$

We have

$\tan^2y=\sec^2y-1\implies x^2=z^2-1\implies z=\sqrt{x^2+1}$

$\implies\cos(\tan^{-1}x)=\cos y=\dfrac1{\sqrt{x^2+1}}$

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Similar strategy: Let $y=\arcsin x$, so that $\sin y=x$. In a right triangle with hypotenuse $z$ and reference angle $y$, we have

$\cos y=\sqrt{1-x^2}[t/ex][tex]\implies\cot(\arcsin x)=\cot y=\dfrac{\cos y}{\sin y}=\dfrac{\sqrt{1-x^2}}x$