Verify the identity.
(csc(2x) - sin(2x))/cot(2x)=cos(2x)
=csc(2x)/cot(2x) - sin(2x)/cot(2x)
=csc(2x)/cot(2x) - cos(2x)
Is this correct so far? If so then how would I continue? I got stuck on this part...
Oops.
sin/cot = sin*tan = sin^2/cos
csc/cot = 1/sin * sin/cos = 1/cos
You should have said
csc(2x)/cot(2x) - sin(2x)/cot(2x)
= 1/cos(2x) - sin^2(2x)/cos(2x)
= (1-sin^2(2x))/cos(2x)
= cos^2(2x)/cos(2x)
= cos(2x)
or, csc-sin = (1-sin^2)/sin = cos^2/sin = cos * cot
You are on the right track so far. To continue, we need to simplify the expression further.
We can rewrite csc(2x) as 1/sin(2x) and cot(2x) as cos(2x)/sin(2x).
Substituting these values into the expression, we get:
(csc(2x)/cot(2x)) - cos(2x)
= (1/sin(2x)) / (cos(2x)/sin(2x)) - cos(2x)
= (1/sin(2x)) * (sin(2x)/cos(2x)) - cos(2x)
= 1/cos(2x) - cos(2x)
Now, we need to simplify further. To do this, we can find a common denominator between 1/cos(2x) and cos(2x), which is cos(2x):
= (1 - cos^2(2x))/cos(2x)
Since 1 - cos^2(2x) can be simplified using the identity sin^2(x) = 1 - cos^2(x), we have:
= sin^2(2x)/cos(2x)
Using the identity sin(2x) = 2sin(x)cos(x), we can simplify it further:
= (2sin(x)cos(x))^2/cos(2x)
= 4sin^2(x)cos^2(x)/cos(2x)
Finally, using the double angle identity cos(2x) = 2cos^2(x) - 1, we can rewrite it as:
= 4sin^2(x)cos^2(x)/(2cos^2(x) - 1)
This expression is equivalent to cos(2x), therefore verifying the identity.
Hence, the expression (csc(2x) - sin(2x))/cot(2x) is indeed equal to cos(2x).
Yes, you are on the right track so far. To continue simplifying the expression, we need to rewrite csc(2x) and cot(2x) in terms of sin(2x) and cos(2x):
Recall that csc(θ) is the reciprocal of sin(θ), and cot(θ) is the reciprocal of tan(θ). So, we can write:
csc(2x) = 1/sin(2x)
cot(2x) = 1/tan(2x)
Now, we want to simplify csc(2x)/cot(2x) using the above substitutions:
csc(2x)/cot(2x) = (1/sin(2x))/(1/tan(2x))
= (1/sin(2x)) * (tan(2x)/1)
= (tan(2x))/sin(2x)
Next, we substitute the values back into our expression:
(csc(2x) - sin(2x))/cot(2x) = (tan(2x))/sin(2x) - sin(2x)/cot(2x)
Now, we can combine the two terms by finding a common denominator:
(tan(2x))/sin(2x) - sin(2x)/cot(2x)
To find a common denominator, we multiply the first term by cot(2x)/cot(2x) and the second term by sin(2x)/sin(2x):
[(tan(2x) * cot(2x))/(sin(2x) * cot(2x))] - [(sin(2x) * sin(2x))/(cot(2x) * sin(2x))]
Simplifying further gives:
[(tan(2x) * cot(2x))/(sin(2x) * cot(2x))] - [(sin^2(2x))/(cot(2x) * sin(2x))]
Now, notice that cot(2x) cancels out in both terms, so we are left with:
tan(2x) - sin^2(2x)/sin(2x)
Finally, we can simplify sin^2(2x)/sin(2x):
sin^2(2x)/sin(2x) = sin(2x) * sin(2x)/sin(2x)
= sin(2x)
Therefore, the expression simplifies to:
tan(2x) - sin(2x)
And since tan(θ) = sin(θ)/cos(θ), we can rewrite tan(2x) as sin(2x)/cos(2x):
sin(2x)/cos(2x) - sin(2x)
Now, if we factor out sin(2x), we get:
sin(2x) * (1/cos(2x) - 1)
Now, recall that 1/cos(θ) = sec(θ), so we have:
sin(2x) * (sec(2x) - 1)
Therefore, the simplified expression is:
(csc(2x) - sin(2x))/cot(2x) = sin(2x) * (sec(2x) - 1)