Trigonometry - half angles problems

[paul18]

1. The problem statement, all variables and given/known data
Previous part of the question I have solved:
Express cos(2x) in terms of sin(x):

I got this answer:
cos(2x)=1-2sin2(x)

Hence or otherwise solve the equation
cos(x) + 3sin(x/2) = 2


2. Relevant equations
Double angle formulae:
cos(2x)=cos(2x) - sin(2x)
sin(2x)=2sin(x)cos(x)


3. The attempt at a solution
So basically I'm doing revision on trig, and I know I've come across this problem before, where you are presented with a full angle and a half angle, but I have failed to find an example. It's something to do with halving a double angle formula I think but I can't even start it. How do I get rid of the half angle? I know I should show an attempt, but I can't even start it off. Just a refresher on how to remove half angles would be great thanks :)
(Yes i have read the rules but this problem is difficult to put into that format properly)

[sjb-2812]

1. The problem statement, all variables and given/known data
Previous part of the question I have solved:
Express cos(2x) in terms of sin(x):

I got this answer:
cos(2x)=1-2sin2(x)

Hence or otherwise solve the equation
cos(x) + 3sin(x/2) = 2


2. Relevant equations
Double angle formulae:
cos(2x)=cos(2x) - sin(2x)
sin(2x)=2sin(x)cos(x)


3. The attempt at a solution
So basically I'm doing revision on trig, and I know I've come across this problem before, where you are presented with a full angle and a half angle, but I have failed to find an example. It's something to do with halving a double angle formula I think but I can't even start it. How do I get rid of the half angle? I know I should show an attempt, but I can't even start it off. Just a refresher on how to remove half angles would be great thanks :)
(Yes i have read the rules but this problem is difficult to put into that format properly)

If the first part read cos(2z) in terms of sin(z), what would your answer be? Then substitute z for x/2 in the second question

[eumyang]

1. The problem statement, all variables and given/known data
Previous part of the question I have solved:
Express cos(2x) in terms of sin(x):

I got this answer:
cos(2x)=1-2sin2(x)

Hence or otherwise solve the equation
cos(x) + 3sin(x/2) = 2


2. Relevant equations
Double angle formulae:
cos(2x)=cos(2x) - sin(2x)
sin(2x)=2sin(x)cos(x)


Note that x = 2 times (x/2). It may be helpful if you substitute another variable, say y, in for x/2. What would x equal? What does this equation:
cos(x) + 3sin(x/2) = 2
look like with the substitutions?

[paul18]

Note that x = 2 times (x/2). It may be helpful if you substitute another variable, say y, in for x/2. What would x equal? What does this equation:
cos(x) + 3sin(x/2) = 2
look like with the substitutions?

cos(2y) + 3sin(y) = 2.
Thanks very much that's what I needed a refresher on :)