I can't solve √(a^2+4) when a= 2 tan θ HELP!!!?

Question

I'm doing my calculus homework and this problem is one of many that i don't understand...could you also help me with these:

2) find all θ in the interval [0,2π) that satisfy the equation sin 2 θ = 0

3) Find all θ in the interval [0,2π) that satisfy the equation

2 cos θ tan θ + tan θ = 0

4) If cos 2θ = 1/3 and 0 ≤ 2θ ≤ π, find cos θ

Answer 1

1)

√[ (2 tan x)² + 4] =

√[ 4 tan² x + 4] =

√[ 4( tan² x + 1)] =

2√(sec² x) =

2| sec x|

sec x could be ≥ 0 or < 0, but sec² x and its square root are both ≥ 0, so we have to use absolute value.

2)

sin 2x = 0

2x = 0 + 2πn, x = 0 + πn, x = 0 or π

and

2x = π + 2πn, x = π/2 + πn, x = π/2 or 3π/2

3)

2 cos x tan x + tan x = 0

(tan x)(2 cos x + 1) = 0

if tan x = 0, x = 0 or x = π

if 2 cos x + 1 = 0, cos x = -½, x = 2π/3 or 5π/3

4)

cos 2x = 1/3

2 cos² x - 1 = 1/3

2 cos² x = 4/3

cos² x = 2/3

cos x = ±(√6)/3