Question

Simplify the expression. csc(-x)/1+tan^2(x)

a. tan(x)
b.-cos(x)tan(x)
c.-cos(x)cot(x)
d.sin(x)tan(x)

Answer 1

`\csc(-x)=\frac1{\sin(-x)}=-\frac1{\sin x}=-\csc x`


`1+\tan^2x=\sec^2x`

So,


`(\csc(-x))/(1+\tan^2x)=-(\csc x)/(\sec^2x)=-(\cos^2x)/(\sin x)=-\cot x\cos x`

So the answer is C.

Answer 2

The simplified expression is:

Option: c

c. -cos(x)cot(x)

Explanation:

We are asked to simplify the expression:

`(\csc (-x))/(1+\tan^2 x)`

We know that :

`\csc (-x)=-\csc x`

Also, we know that:

`\sec^2 x-\tan^2 x=1\\\\i.e.\\\\\sec^2 x=1+\tan^2 x`

i.e.

`(\csc (-x))/(1+\tan^2 x)=(-\csc x)/(\sec^2 x)`

Also, we know that:

`\csc x=(1)/(\sin x)`

and

`\sec x=(1)/(\cos x)\\\\\\i.e.\\\\\\\sec^2 x=(1)/(\cos^2 x)\\\\i.e.\\\\\cos^2x=(1)/(\sec^2 x)`

Hence, we have:

`(\csc (-x))/(1+\tan^2 x)=-(\cos^2 x)/(\sin x)`

which could also be written as:

`(\csc (-x))/(1+\tan^2 x)=-(\cos x)/(\sin x)* \cos x`

Now, we have:

`\cot x=(\cos x)/(\sin x)`

Hence, we get:

`(\csc (-x))/(1+\tan^2 x)=-\cos x\cot x`