Sec^2(u/2)=(2sec(u))/(sec(u)+1)
It would be helpful to know what you need with this identity. Prove it? One wonders.
If so, you need to start on another identity.
Remember cos(2x)=cos^2(x) - sin^2(x)=cos^2(x)-(1-cos^2(x))=2cos^2(x)-1 or
cos^2(x)=1/2 (1+cos(2x))
now replace 2x=u and you have
cos^2(u/2)=1/2 (1+cos(u))
and
sec^2(u/2)=2/(1+1/sec(u))
now multiply the right side by secu/secu
sec^2(u/2)=2sec(u)/(sec(u)+1)
viola.
To prove that the equation sec^2(u/2) = (2sec(u))/(sec(u)+1) is true, we'll start by simplifying both sides individually.
Let's start with the left-hand side (LHS):
sec^2(u/2)
Using the identity, sec^2(x) = 1/cos^2(x), we can rewrite sec(u/2) as 1/cos^2(u/2):
1/cos^2(u/2)
Next, we'll simplify the right-hand side (RHS):
(2sec(u))/(sec(u)+1)
Using the identity, sec(x) = 1/cos(x), we can rewrite sec(u) as 1/cos(u):
(2(1/cos(u)))/(1/cos(u)+1)
Next, we'll simplify the RHS further:
(2/cos(u))/((1+cos(u))/cos(u))
Simplifying the expression inside the denominator:
(2/cos(u))/(1/cos(u)*(1+cos(u)))
Now, we'll cancel out the common factor of cos(u):
2/(1+cos(u))
Now, let's compare the LHS and RHS:
LHS: 1/cos^2(u/2)
RHS: 2/(1+cos(u))
To demonstrate that these two expressions are equal, we need to manipulate one to become the other.
To do this, we'll use the identity, cos^2(x) = (1+cos(2x))/2:
1/cos^2(u/2) = 2/(1+cos(u))
So, by using the identity cos^2(x) = (1+cos(2x))/2, we have proven that sec^2(u/2) = (2sec(u))/(sec(u)+1) is true.
To solve the equation sec^2(u/2) = (2sec(u))/(sec(u)+1), we can start by manipulating the equation to simplify it.
Using the identity sec^2(u) = 1 + tan^2(u), we can rewrite sec^2(u/2) as (1 + tan^2(u/2)). Similarly, we can rewrite sec(u) as 1/cos(u).
Now, let's substitute these values into the equation:
(1 + tan^2(u/2)) = (2 * (1/cos(u))) / (1/cos(u) + 1)
Next, we can simplify the equation further by canceling out some terms:
1 + tan^2(u/2) = 2 / (1 + cos(u))
Now, let's eliminate the fractions by multiplying both sides of the equation by (1 + cos(u)):
(1 + cos(u))(1 + tan^2(u/2)) = 2
Expanding the left side of the equation:
1 + cos(u) + tan^2(u/2) + cos(u) * tan^2(u/2) = 2
Now, we can rewrite tan^2(u/2) as (sin^2(u/2) / cos^2(u/2)) using the identity tan^2(u/2) = sin^2(u/2) / cos^2(u/2):
1 + cos(u) + sin^2(u/2) / cos^2(u/2) + cos(u) * (sin^2(u/2) / cos^2(u/2)) = 2
Multiplying through by cos^2(u/2) to eliminate the fractions:
cos^2(u/2) + cos(u) * sin^2(u/2) + sin^2(u/2) + cos(u) * sin^2(u/2) = 2 * cos^2(u/2)
Simplifying the equation further:
cos^2(u/2) + 2 * cos(u) * sin^2(u/2) + 2 * sin^2(u/2) = 2 * cos^2(u/2)
Combining like terms:
cos^2(u/2) + 2 * sin^2(u/2) + 2 * cos(u) * sin^2(u/2) = 2 * cos^2(u/2)
Subtracting 2 * cos^2(u/2) from both sides:
cos^2(u/2) + 2 * sin^2(u/2) + 2 * cos(u) * sin^2(u/2) - 2 * cos^2(u/2) = 0
Simplifying further:
2 * sin^2(u/2) + 2 * cos(u) * sin^2(u/2) = 0
Factoring out 2 * sin^2(u/2):
2 * sin^2(u/2) * (1 + cos(u)) = 0
Now, there are two cases to consider:
Case 1: sin^2(u/2) = 0
This implies that u/2 = 0, which means u = 0.
Case 2: 1 + cos(u) = 0
This implies that cos(u) = -1, which means u = (2n + 1) * π, where n is an integer.
So, the solution to the original equation sec^2(u/2) = (2sec(u))/(sec(u)+1) is u = 0 or u = (2n + 1) * π, where n is an integer.