There are a few questions that my teacher did not make clear when explaining our lesson. Can you guys explain?

1. How could these be simplified?

a. 1+tan² theta / 1+cot theta

I already know that it simplifies to:

sec² theta / csc² theta

What else could be done?

No idea about the rest...

b. sect+csct / 1+tant

c. sin(x+y)cosy - cos(x+y)siny

d. 2sin2asin2a

e. Square root of: 1+cos2beta / 2 (end square root)

Restrictions: -pi/2 < beta < pi/2

f. Dear god this is a hard one.

tan7pi/6 - tanpi/3 all over 1 + tan7pi/6 tanpi / 3

I promise, you guys aren't doing my homework. :P I just have a bad teacher.

2. Solve the trig functions within [0, 2pi)

a. cos3x + cosx = 0

b. cot²x + cos²x = cot²xcos²x

3. Stupid proofs... Prove the identities. T__T'

a. cscx/cscx-1 + cscx/cscx+1 = 2sec²x

b. sinasinbeta(cotacotbeta-1) = cos(a+beta)

c. cos^4y - sin^4y = cos2y

d. tan(pi/4 + p)

4. Find the values (w/out calculator)

a. cos[cos^-1(-1/2) + sin^-1 1]

b. tan[cos^-1 (-1)-cos^-1 1]

5. :/ Find exact values of sin2u and cos u/2.

a. cscu = 2

Restriction: pi/2 < u < pi

b. tanu = -8/5

Restriction: 3pi/2 < u < 2pi

I'm soo sorry. I feel like I'm asking way too much. x___x'