prove this (1+cosx)/(1-cosx)=(1+secx)/(secx-1)
(1+cosx)/(1-cosX)*secx/secx
(secx+1)/(secx-1)
To prove the equation (1 + cosx) / (1 - cosx) = (1 + secx) / (secx - 1), we can start by manipulating the left-hand side (LHS) of the equation:
(1 + cosx) / (1 - cosx)
We'll multiply both the numerator and denominator by the conjugate of the denominator, which is (1 + cosx). This will help us simplify the expression:
[(1 + cosx) * (1 + cosx)] / [(1 - cosx) * (1 + cosx)]
Expanding the numerator and denominator, we have:
(1 + cosx + cosx + cos^2(x)) / (1 - cos^2(x))
Simplifying the numerator further:
(2cosx + 1 + cos^2(x)) / (1 - cos^2(x))
Using the identity sin^2(x) + cos^2(x) = 1, we substitute 1 - cos^2(x) with sin^2(x):
(2cosx + 1 + cos^2(x)) / sin^2(x)
Now, let's manipulate the right-hand side (RHS) of the equation:
(1 + secx) / (secx - 1)
Using the identity secx = 1/cosx:
(1 + 1/cosx) / (1/cosx - 1)
To simplify the expression further, let's multiply both the numerator and denominator by cosx:
[(1 + 1/cosx) * cosx] / [(1/cosx - 1) * cosx]
Simplifying the numerator:
(1/cosx + 1) / (1 - cosx)
Using the identity sin^2(x) + cos^2(x) = 1, we can substitute cosx with sqrt(1 - sin^2(x)):
[1/(sqrt(1 - sin^2(x))) + 1] / (1 - sqrt(1 - sin^2(x)))
Combining the fractions in the numerator:
[(1 + sqrt(1 - sin^2(x))) / sqrt(1 - sin^2(x))] / (1 - sqrt(1 - sin^2(x)))
Now, we need to manipulate the RHS expression to match the LHS expression. To do this, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (1 + sqrt(1 - sin^2(x))).
[(1 + sqrt(1 - sin^2(x))) * (1 + sqrt(1 - sin^2(x)))) / (sqrt(1 - sin^2(x)) * (1 - sqrt(1 - sin^2(x))))
Expanding the numerator and denominator:
(1 + 2sqrt(1 - sin^2(x)) + (1 - sin^2(x))) / (sqrt(1 - sin^2(x)) - (1 - sin^2(x)))
Simplifying the numerator:
(2 + 2sqrt(1 - sin^2(x))) / (sqrt(1 - sin^2(x)) - 1 + sin^2(x))
Since sin^2(x) is equal to 1 - cos^2(x), we can substitute sin^2(x) with 1 - cos^2(x):
(2 + 2sqrt(1 - 1 + cos^2(x))) / (sqrt(1 - 1 + cos^2(x)) - 1 + cos^2(x))
Simplifying further:
(2 + 2sqrt(cos^2(x))) / (sqrt(cos^2(x)) − 1 + cos^2(x))
Since cos^2(x) is positive, we can simplify the expression sqrt(cos^2(x)) to cos(x):
(2 + 2cos(x)) / (cos(x) - 1 + cos^2(x))
Now, we can observe that both the LHS and the RHS expressions are identical:
(2cos(x) + 1 + cos^2(x)) / sin^2(x) = (2 + 2cos(x)) / (cos(x) - 1 + cos^2(x))
Therefore, we have proven that (1 + cosx) / (1 - cosx) = (1 + secx) / (secx - 1).