How can I do the following questions?

How do you find the value of #tan(67.5˚)#

How do you prove that #costheta/(1 + sin theta)  costheta/(1+sintheta) = 2tantheta# ?

How do you find the value of
#tan(67.5˚)# 
How do you prove that
#costheta/(1 + sin theta)  costheta/(1+sintheta) = 2tantheta# ?
2 Answers
Start by putting the left side on a common denominator.
Explanation:
Use the pythagorean identity
Now apply the quotient identity
Identity proved!
Hopefully this helps!
Solving number 30...
Explanation:
Note that
Looking at the given identity:
#tan(2theta)=(2tan(theta))/(1tan^2(theta))#
If we let
#tan(2xx67 1/2˚)=(2tan(67 1/2˚))/(1tan^2(67 1/2˚))#
#tan(135˚)=(2tan(67 1/2˚))/(1tan^2(67 1/2˚))#
We already know the value of
#1=(2tan(67 1/2˚))/(1tan^2(67 1/2˚))#
This will be easier to look at if we let
#1=(2u)/(1u^2)#
Crossmultiply. We want to solve for
#1(1u^2)=2u#
#u^21=2u#
Solve like you normally would a quadratic equation (set it equal to
#u^22u1=0#
You could use the quadratic formula here, but I'll complete the square:
#u^22u=1#
We want the left side to match
#u^22u+1=1+1#
#(u1)^2=2#
Take the square root of both sides:
#u1=+sqrt2#
#u=1+sqrt2#
Since
#tan(67 1/2˚)=1+sqrt2#
However, something's up... what should we do about the plus or minus sign? The tangent of a single angle can only equal one thing.
Since
#tan(67 1/2˚)=1+sqrt2#