How would you establish this identity:
(1+sec(beta))/(sec(beta))=(sin^2(beta))/(1-cos(beta))
on the right, sin^2 = 1-cos^2, that factor to 1-cos * `1+cos, then the denominator makes the entire right side 1+cosB
which is 1+1/sec which is 1/sec (sec+1)
qed
using sec(beta) = 1/cos(beta):
1+sec(beta))/(sec(beta))= 1 + cos(beta)
sin^2(beta)/(1-cos(beta)) =
(1-cos^2(beta))/(1-cos(beta)) =
1 + cos(beta)
This follows e.g. from:
(1 - x^2) = (1 - x)(1 + x)
and thus:
(1 - x^2)/(1 - x) = 1 + x
x=(-1)
To establish the given identity, we can start by simplifying both sides separately.
On the left side of the equation, we have (1 + sec(beta))/(sec(beta)). Since sec(beta) is equal to 1/cos(beta), we can substitute it in the equation:
(1 + 1/cos(beta))/(1/cos(beta))
Now, we can simplify the expression by multiplying the numerator and denominator by cos(beta) to get rid of the fraction:
[(1 + 1/cos(beta))*(cos(beta))]/[(1/cos(beta))*(cos(beta))]
This simplifies to:
(cos(beta) + 1)/(1)
Which is just cos(beta) + 1.
On the right side of the equation, we have (sin^2(beta))/(1-cos(beta)). Using the identity sin^2(beta) = 1-cos^2(beta), we can substitute it in the equation:
[(1 - cos^2(beta))/(1 - cos(beta))]
This expression can be further simplified using the difference of squares identity, which states that (a^2 - b^2) = (a - b)(a + b):
[(1 - cos(beta))(1 + cos(beta))]/[(1 - cos(beta))]
The (1 - cos(beta)) term simplifies to 1:
1 + cos(beta)
So, the right side of the equation simplifies to 1 + cos(beta).
Comparing the simplified expressions on both sides, we can see that:
Left side: cos(beta) + 1
Right side: 1 + cos(beta)
Since both sides of the equation are equal, the identity is established.