I need help with some trig equalities homework problems

Question

how do i prove cot(x/2) = (1 cosx)/sinx

and

solve for x in the equation cos(^2)x = .5sin2x

Answer 1

Let s = (1 + cos(x)) / sin(x).

Using the half angle formulae:

s = [ 2 cos^2(x / 2) ] / [ 2 sin(x / 2) cos(x / 2) ]

= cos(x / 2) / sin(x / 2)

= cot(x / 2).

2.

cos^2(x) = 0.5 sin(2x)

= 0.5 * 2 sin(x) cos(x)

= sin(x) cos(x)

cos^2(x) - sin(x) cos(x) = 0

cos(x) ( cos(x) - sin(x) ) = 0

cos(x) = 0 gives x = pi / 2, 3pi / 2, 5pi / 2 ...

cos(x) - sin(x) = 0 gives:

cos(x) = sin(x)

tan(x) = 1

x = pi / 4, 5pi / 4, 9pi / 4, 13pi / 4 ...