How do you simplify the expression $tan (arcsin (\frac{x}{5}))$ $tan(arcsin(x/5))$ ?

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Answer 1 · 2016-08-01T12:05:57

$\frac{x}{\sqrt{25 - x^{2}}}$ $x/sqrt(25-x^2)$

Let $a = a r c sin (\frac{x}{5}) \in [- \frac{π}{2}, \frac{π}{2}]$ $a = arc sin (x/5) in [-pi/2, pi/2]$

Then, $sin a = \frac{x}{5} \in [- 1, 1]$ $sin a =x/5 in [-1, 1]$ , and so, $x \in [- 5, 5]$ $x in [-5, 5]$ ,

Now, the given expression is

$tan a$ $tan a$

$= \frac{sin a}{cos a}$ $= sin a/cos a$

$= \frac{sin a}{\sqrt{1 - {sin}^{2} a}}$ $=sin a/sqrt(1-sin^2a)$

$= \frac{\frac{x}{5}}{\sqrt{1 - {(\frac{x}{5})}^{2}}}$ $=(x/5)/sqrt(1-(x/5)^2)$
$= \frac{x}{\sqrt{25 - x^{2}}}$ $=x/sqrt(25-x^2)$

How do you simplify the expression tan(arcsin(x5))tan(arcsin(x/5)) ?