You're asked to simplify
the given trig expressions.
To simplify the expressions,
we will first perform a substitution
using one of the Pythagorean identities
and then convert to sines and cosines.
For the first expression we have
one plus cotangent squared x
over cosecant squared x minus one.
Looking at the identities below,
since cotangent squared theta plus one
equals cosecant squared theta,
we can substitute cosecant squared x
for one plus cotangent squared x.
Using the same identity, notice how,
if we subtract one on both sides,
we have cotangent squared theta equals
cosecant squared theta minus one,
and therefore we can substitute cotangent squared x
for cosecant squared x minus one.
And now let's convert to sines and cosines.
Since cosecant theta is equal to one over sine theta,
cosecant squared x is equal to one over sine squared x.
And cotangent theta is equal to cosine theta
divided by sine theta
and therefore cotangent squared x
is equal to cosine squared x divided by sine squared x.
And now remember this fraction bar represents division
and dividing by a fraction is equivalent
to multiplying by the reciprocal.
And therefore, let's write this quotient as a product
which will give us one over sine squared x
times the reciprocal of cosine squared x over sine squared x
which is sine squared x
over cosine squared x.
Now before multiplying, we simplify out the common factors
between the numerators and denominators.
Notice here we have a common factor of sine squared x
and therefore these common factors simplify to one.
Multiplying we have one over cosine squared x.
And since one over cosine theta is equal to secant theta,
one over cosine squared x equals secant squared x.
The given expression simplifies to secant squared x.
And now let's take a look at the second expression.
Since tan squared theta plus one equals secant squared theta
we can substitute secant squared x
for tan squared x plus one.
Now looking at the denominator
we have one minus secant squared x.
Well if we know tan squared x
plus one equals secant squared x
and we have one minus secant squared x,
let's subtract secant squared x on both sides
as well as subtract tan squared x on both sides.
This gives us one minus secant squared x
equals negative tangent squared x.
So we can substitute negative tan squared x
for one minus secant squared x.
And now we'll convert to sines and cosines.
Secant squared x is equal to one over cosine squared x
and negative tangent squared x is equal to
negative sine squared x divided by cosine squared x.
And again from here, because this fraction bar
represents division,
instead of dividing, we'll multiply by the reciprocal.
So as a product we have one over cosine squared x
times the reciprocal of negative sine squared x
over cosine squared x, which is negative cosine squared x
over sine squared x.
And now simplifying before multiplying,
we have a common factor of cosine squared x
between the numerators and denominators here and here
that simplify to one.
But be careful here, this simplifies to negative one.
So multiplying we have negative one over sine squared x.
And since one over sine x is equal cosecant x,
this is equal to negative cosecant squared x.
The given expression simplifies to
negative cosecant squared x.
I hope you found this helpful.