Rewrite tan^2x sin^2x in terms of cosine without power greater than one.
Rewrite cot^2〖x/4〗 in terms of cosine without power greater than one. Assume that x is in QI.
tan^2x sin^2x
(sec^2x - 1)(1-cos^2x)
sec^2x - 1 - 1 + cos^2x
1/cos^2x - 2 + cos^2x
(1/cosx - cosx)^2
God luck with that. Or use your double-angle formula a couple of times.
tan^2x sin^2x
cos^2x = (1 + cos2x)/2
sin^2x = (1 - cos2x)/2
You wind up with
(cos4x - 4cos2x + 3) / 4(cos2x + 1)
Use your half-angle formulas for the other one.
tan^2x sin^2x
= (sin^2 x)(sin^2 x)/cos^2x
= (1-cos^2 x)(1 - cos^2 x)/cos^2 x
= (1 - 2cos^2 x + cos^4 x)/cos^2x
= 1/cos^2 x - 2 +cos^2 x
now remember that cos (2A) = 2cos^2 A - 1
which leads to cos^2 A = (cos (2A) + 1)/2
so above
= 2/(cos (2x) + 1) - 2 + (cos (2x) + 1)/2
which contains only cosines and the highest degree is 1
For the 2nd, cot^2 (x/4)
= cos^2 (x/4) / sin^2 (x/4)
use the same as above after you replace the sin^2 with (1 - cos^2 )
To rewrite tan^2(x) sin^2(x) in terms of cosine without a power greater than one, we can use the following trigonometric identity:
sin^2(x) = 1 - cos^2(x)
Using this identity, we can rewrite tan^2(x) sin^2(x) as:
tan^2(x) sin^2(x) = tan^2(x) (1 - cos^2(x))
Now, let's rewrite cot^2(x/4) in terms of cosine without a power greater than one. Since x is in QI, which means 0 < x < π/2, we can use the following trigonometric identity:
cot(x) = 1 / tan(x)
Using this identity, we can rewrite cot^2(x/4) as:
cot^2(x/4) = (1/tan(x/4))^2
Now, we know that tan(x/4) = sin(x/4) / cos(x/4), therefore:
cot^2(x/4) = (1 / (sin(x/4) / cos(x/4)))^2
Simplifying this expression further, we can rewrite cot^2(x/4) as:
cot^2(x/4) = (cos(x/4) / sin(x/4))^2
Therefore, tan^2(x) sin^2(x) can be rewritten as tan^2(x) (1 - cos^2(x)), and cot^2(x/4) can be rewritten as (cos(x/4) / sin(x/4))^2.
To rewrite tan^2x sin^2x in terms of cosine without powers greater than one, we can start by using the trigonometric identity:
sin^2x = 1 - cos^2x
Using this identity, we can rewrite tan^2x sin^2x as:
tan^2x sin^2x = tan^2x(1 - cos^2x)
Next, we rewrite tan^2x in terms of sine and cosine by using another trigonometric identity:
tan^2x = sin^2x / cos^2x
Substituting this into the previous equation, we get:
tan^2x sin^2x = (sin^2x / cos^2x)(1 - cos^2x)
Simplifying further:
= sin^2x - sin^2x cos^2x
= sin^2x(1 - cos^2x)
Now, we have rewritten tan^2x sin^2x in terms of cosine without powers greater than one.
Regarding cot^2(x/4), we can rewrite it in terms of cosine without powers greater than one using the same approach.
First, let's rewrite cot^2(x/4) in terms of sine and cosine:
cot^2(x/4) = (cos^2(x/4) / sin^2(x/4))
Next, we can rewrite cos^2(x/4) in terms of cosine without powers greater than one using the identity:
cos^2(x/4) = 1 - sin^2(x/4)
Substituting this back into the previous equation, we get:
cot^2(x/4) = [(1 - sin^2(x/4)) / sin^2(x/4)]
Simplifying:
= (1 / sin^2(x/4)) - 1
= csc^2(x/4) - 1
Therefore, cot^2(x/4) can be rewritten in terms of cosine without powers greater than one as csc^2(x/4) - 1.