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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)

User Zalew
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2 Answers

5 votes

\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.
User Salexch
by
9.1k points
1 vote

Answer:

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

User Wezten
by
8.3k points

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