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=(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=(1)/(\sin \theta)`

From the triangle,

`\sin \theta=(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}`