verify the identity:
sec(beta)+ tan (beta)= cos(beta)/ 1-sin(beta)
The left side can be rewritten
= (1 + sin theta]/cos theta
Crossmultiply
cos^2 theta = 1 - sin^2 theta
That is an identity
To verify the identity sec(beta) + tan(beta) = cos(beta) / (1 - sin(beta)), we will manipulate the left-hand side (LHS) and right-hand side (RHS) of the equation separately to see if they become equal.
Starting with the LHS:
sec(beta) + tan(beta)
Using the definitions of secant and tangent:
1 / cos(beta) + sin(beta) / cos(beta)
To simplify, we can find a common denominator:
(1 + sin(beta)) / cos(beta)
Now, let's simplify the RHS:
cos(beta) / (1 - sin(beta))
To simplify further, we can multiply both the numerator and denominator by (1 + sin(beta)):
[cos(beta) * (1 + sin(beta))] / [(1 - sin(beta)) * (1 + sin(beta))]
Expanding the denominator:
cos(beta) + cos(beta) * sin(beta) / (1 - sin^2(beta))
Using the identity sin^2(beta) + cos^2(beta) = 1:
cos(beta) + cos(beta) * sin(beta) / cos^2(beta)
Canceling out the common factor of cos(beta):
1 + sin(beta) / cos(beta)
Now, we can observe that both the LHS and RHS have simplified to the same expression:
(1 + sin(beta)) / cos(beta)
Hence, by the process of simplification, we have verified that sec(beta) + tan(beta) = cos(beta) / (1 - sin(beta)).
Note: It's important to note that the validity of this identity depends on the given conditions. For example, if beta is an angle for which the denominators become zero, then the identity may not hold true.