Verify the identity.
(secx + tanx)/(secx - tanx) = (1 + 2sinx + sin(^2)x)/cos(^2)x
To verify the given identity, we need to simplify both sides of the equation and show that they are equal.
Starting with the left-hand side (LHS):
(sec(x) + tan(x))/(sec(x) - tan(x))
To simplify this, we can use the trigonometric identity:
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)
Substituting the above identities into the LHS expression:
((1/cos(x)) + (sin(x)/cos(x))) / ((1/cos(x)) - (sin(x)/cos(x)))
Now, we need to combine the fractions in the numerator and denominator:
[(1 + sin(x)) / cos(x)] / [(1 - sin(x)) / cos(x)]
Next, we can simplify further by multiplying the numerator and denominator by the reciprocal of the denominator:
[(1 + sin(x)) / cos(x)] * [cos(x) / (1 - sin(x))]
Now, we can cancel out the common terms of cos(x) in the numerator and denominator:
(1 + sin(x)) / (1 - sin(x))
Now, let's tackle the right-hand side (RHS):
(1 + 2sin(x) + sin^2(x)) / cos^2(x)
Next, we can simplify the RHS by dividing both numerator and denominator by cos^2(x):
[(1 + 2sin(x) + sin^2(x)) / cos^2(x)] * [1/cos^2(x)]
Now, we can rewrite sin^2(x) using the identity:
sin^2(x) = 1 - cos^2(x)
[(1 + 2sin(x) + (1 - cos^2(x))) / cos^2(x)] * [1/cos^2(x)]
Simplifying further:
[(2 + 2sin(x) - cos^2(x)) / cos^2(x)] * [1/cos^2(x)]
Next, we can distribute the numerator:
[2 + 2sin(x) - cos^2(x)] / cos^2(x)
Now, we can rewrite cos^2(x) as 1 - sin^2(x):
[2 + 2sin(x) - (1 - sin^2(x))] / cos^2(x)
Simplifying further:
[2 + 2sin(x) - 1 + sin^2(x)] / cos^2(x)
Combining like terms:
[1 + 2sin(x) + sin^2(x)] / cos^2(x)
Now, we can see that the simplified left-hand side and right-hand side of the equation are equal:
(1 + sin(x)) / (1 - sin(x)) = [1 + 2sin(x) + sin^2(x)] / cos^2(x)
Hence, the identity has been verified.