Answer:
Opt. 4: Tan(15)=tan(30/2)= +/- sqrt(1-cos30) / sqrt(1+cos(30))
Step-by-step explanation:
To solve this identity is a key objective to remind the following expression:
(sin(x/2))^2= 1 - (cos(x/2))^2 (a)
The following to remember is this:
cos(x)=cos(x/2 + x/2)= (cos(x/2))^2 - (sin(x/2))^2 (b)
Lets substitute equation (a) in (b)
we have (cos(x/2))^2= (1+cos(x))/2
Lets apply square root to both sides of equality sign:
sqrt((cos(x/2))^2= (1+cos(x))/2)
cos(x/2)= sqrt( (1+cos(x))/2 ) (c)
Use (c) in (a)
(sin(x/2))^2 = 1 - (1 + cos(x)) /2
Lets apply square root to both sides of equality sign:
sqrt((sin(x/2))^2 = 1 - (1 + cos(x)) /2)
We obtain that:
sin(x/2)= sqrt (1-cos(x) ) /2 ) (d)
Finally tan(x/2) is equation (d) divided by equation (c)
[tex]tan(x/2) = \sqrt{\frac{1-cos(x)}{1+cos(x)}}[/tex]
Just need to evaluate x=30.
The signs +/- comes out sinces there is a square root involved and you can have either a positive answer or a negative.
