Math Help - Inverse Trig funtions...

1. Inverse Trig funtions...

Write:

Tan(Tan^(-1) v + Sin ^(-1) v)

in terms of V, with no trig functions or inverse trig functions. How do I do this?

Also, if I have cos(2sin^(-1) X), I would set (2sin^(-1) X) as A, and say that 2 Sin A=X (and thus, SinA=X/2), correct?

Thanks!

2. Originally Posted by JasonW
Write:

Tan(Tan^(-1) v + Sin ^(-1) v)

in terms of V, with no trig functions or inverse trig functions. How do I do this?

Also, if I have cos(2sin^(-1) X), I would set (2sin^(-1) X) as A, and say that 2 Sin A=X (and thus, SinA=X/2), correct?

Thanks!
Not sure about the first one, but for the second one, you don't need to introduce any new variables.

$cos(2sin^{-1}x)$

Use double angle formula:
$cos^{2}(sin^{-1}x)-sin^{2}(sin^{-1}x)$

Now draw a triangle, you know that the inverse sin of x is an angle, and the sine of that angle is x/1, so label the opposite side x, and the hypotenuse 1, then by the Pythagorean theorem, the adjacent side is $\sqrt{1-x^2}$. So that is your triangle and the values for each side.

So the cosine of that angle will be the adjacent side over the hypotenuse, and the sine of that angle will be the opposite side over the hypotenuse. Now fill in the values.

$(\frac{\sqrt{1-x^{2}}}{1})^{2}-(\frac{x}{1})^{2}$

Simplify
$(\sqrt{1-x^{2}})^{2}-(x)^{2}$

Simplify
$1-x^{2}-x^{2}$

Simplify
$1-2x^{2}$

So:
$cos(2sin^{-1}x)=1-2x^{2}$

3. Originally Posted by JasonW
Write:

Tan(Tan^(-1) v + Sin ^(-1) v)

in terms of V, with no trig functions or inverse trig functions. How do I do this?
By the addition formula for tangent, we have:

$\tan \left( \tan^{-1} v + \sin^{-1} v \right) = \frac {\tan \left( \tan^{-1} v \right) + \tan \left( \sin^{-1} v \right)}{1 - \tan \left( \tan^{-1} v \right) \tan \left( \sin^{-1} v \right)}$

Now apply angel.white's method to simplify $\tan \left( \tan^{-1} v \right)$ and $\tan \left( \sin^{-1} v \right)$

(it's safe to assume $\tan \left( \tan^{-1} v \right) = v$, we don't have a problem with the range of tangent like we do with other inverse trig functions)