Question

Simplify the given expression:csc(tan^-1)(w)=

Answer 1

The problem is given to be:

$\csc (\tan ^{-1}(w))$

Let

$\begin{gathered} \theta=\tan ^{-1}w \\ \therefore \\ \tan \theta=w \end{gathered}$

We can write the above to be:

$\tan \theta=\frac{w}{1}$

Using the above, we can draw a right-angled triangle as shown below:

To find the value of x, we can use the Pythagorean Theorem:

$\begin{gathered} x^2=w^2+1^2 \\ x=\sqrt[]{w^2+1} \end{gathered}$

Recall:

$\csc (\tan ^{-1}(w))=\csc \theta$

The identity of cosec is given to be:

$\csc \theta=\frac{1}{\sin \theta}$

From the triangle,

$\sin \theta=\frac{w}{x}=\frac{w}{\sqrt[]{w^2+1}}$

Therefore,

$\csc \theta=\frac{\sqrt[]{w^2+1}}{w}$

Therefore, the answer is given to be:

$\csc (\tan ^{-1}(w))=\frac{\sqrt[]{w^2+1}}{w}$