Question

Simplify the expression. tan(sin^−1 x)

Answer 1

looking at that picture
we could use the pythagorean theorem to find the adjacent side,
or cosine value
$\bf tan\left[ sin^{-1}(x) \right]\implies \begin{cases} x=sin(\theta)\\ or\\ \cfrac{x}{1}=sin(\theta)\implies \cfrac{x=opposite=b}{1=radius=c} \end{cases}\\ \\ \quad \\ c^2=a^2+b^2\implies \sqrt{c^2-b^2}=a \\ \quad \\ \sqrt{1^2-x^2}=a=cos(\theta)\\ ----------------------------\\ thus \\ \quad \\ tan\left[ sin^{-1}(x) \right]\implies tan(\theta)\implies \cfrac{sin(\theta)}{cos(\theta)}\implies \cfrac{\frac{x}{1}}{\sqrt{1-x^2}} \\ \quad \\ \cfrac{x}{\sqrt{1-x^2}}$

Answer 2

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Let  $\mathsf{\theta=sin^{-1}(x)\qquad\qquad-\dfrac{\pi}{2}\ \textless \ \theta\ \textless \ \dfrac{\pi}{2}.}$

(that is the range of the inverse sine function).


So,

$\mathsf{sin\,\theta=sin\!\left[sin^{-1}(x)\right]}\\\\ \mathsf{sin\,\theta=x\qquad\quad(i)}$


Square both sides:

$\mathsf{sin^2\,\theta=x^2\qquad\qquad(but~sin^2\,\theta=1-cos^2\,\theta)}\\\\ \mathsf{1-cos^2\,\theta=x^2}\\\\ \mathsf{1-x^2=cos^2\,\theta}\\\\ \mathsf{cos^2\,\theta=1-x^2}$


Since $\mathsf{-\,\dfrac{\pi}{2}\ \textless \ \theta\ \textless \ \dfrac{\pi}{2},}$ then $\mathsf{cos\,\theta}$ is positive. So take the positive square root and you get

$\mathsf{cos\,\theta=\sqrt{1-x^2}\qquad\quad(ii)}$


Then,

$\mathsf{tan\,\theta=\dfrac{sin\,\theta}{cos\,\theta}}\\\\\\ \mathsf{tan\,\theta=\dfrac{x}{\sqrt{1-x^2}}}\\\\\\\\ \therefore~~\mathsf{tan\!\left[sin^{-1}(x)\right]=\dfrac{x}{\sqrt{1-x^2}}\qquad\qquad -1\ \textless \ x\ \textless \ 1.}$


I hope this helps. =)


Tags:  inverse trigonometric function sin tan arcsin trigonometry