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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
rahul
answered11 years ago
Explain Bot
answered11 months ago
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.