Question

Rewrite tan^2⁡x sin^2⁡x 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.

Answer 1

tan^2⁡x sin^2⁡x
= (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 )

Answer 2

tan^2⁡x sin^2⁡x
(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^2⁡x sin^2⁡x
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.