1. ## inverse trigonometric functions

Help needed on these two problems -

1. Prove that

tan^(-1) x = sin^(-1)( x / sqrt(1 + x^2))

2. Find the exact value of -
$\displaystyle \cos\left( \tan^{-1}(\frac {5}{12}) + \cot^{-1}(\frac {5}{12})\right)$

2. Originally Posted by darkangel
Help needed on these two problems -

1. Prove that

tan^(-1) x = sin^(-1)( x / sqrt(1 + x^2))

2. Find the exact value of -
$\displaystyle \cos\left( \tan^{-1}(\frac {5}{12}) + \cot^{-1}(\frac {5}{12})\right)$
for the second one rewrite it as $\displaystyle cos\bigg(arctan\bigg(\frac{5}{12}\bigg)+arctan\big g(\frac{12}{5}\bigg)\bigg)$..then split it apart using $\displaystyle cos(x+y)=cos(x)\cdot{cos(y)}+sin(x)\cdot{sin(y)}$ and use the fact that $\displaystyle cos(arctan(x))=\frac{1}{\sqrt{1+x^2}}$ to finish

3. The second one should be easy.

$\displaystyle tan^{-1}(x)+cot^{-1}(x)=\frac{\pi}{2}$

4. 1. Set $\displaystyle y = \sin^{-1} \frac{x}{\sqrt{1+x^{2}}}$. This implies:

$\displaystyle \sin y = \frac{x}{\sqrt{1+x^{2}}}$

Using the identity: $\displaystyle \sin^{2} x + \cos^{2} x = 1 \Rightarrow \cos x = \sqrt{1 - \sin^{2} x}$

So, find an expression of $\displaystyle \cos y$.

Then, find tan y using the relation: $\displaystyle \tan y = \frac{\sin y}{\cos y}$ and solve for y.

5. aite Thanks for everyone's help

6. Originally Posted by galactus
$\displaystyle tan^{-1}(x)+cot^{-1}(x)=\frac{\pi}{2}$
Just a quick note here.

Observe that $\displaystyle \left( {\tan ^{ - 1} x} \right)' + \left( {\cot ^{ - 1} x} \right)' = 0,$ so $\displaystyle \tan ^{ - 1} x - \frac{\pi } {4} + \cot ^{ - 1} x - \frac{\pi } {4} = 0,$ and the result follows.

7. woah ! thanks Krizalid !!