We have an answer to "Simplify tan⁡(sin−1(x))"

Roger Smith

Answered question

2021-12-16

Simplify

\tan (\sin^{- 1} (x))

Ben Owens

Beginner2021-12-17Added 27 answers

Let

\sin^{- 1} (x) = θ

, then

x = \sin θ

If we draw a right triangle with hypotenuse equal to 1 and the other side equals to x.
From pythagorean theorem:

\sqrt{1 - x^{2}}

is the other side.
Thus,

\tan θ = \frac{\sin θ}{\cos θ} = \frac{\sin θ}{\sqrt{1 - \sin^{2} θ}}

x = \sin θ

Now, we have:

\tan θ = \frac{x}{\sqrt{1 - x^{2}}}

From

\sin^{- 1} (x) = θ

we get

\tan (\sin^{- 1} (x)) = \frac{x}{\sqrt{1 - x^{2}}}

Terry Ray

Beginner2021-12-18Added 50 answers

\tan (\arcsin (x)) = \frac{x \sqrt{1 - x^{2}}}{1 - x^{2}}

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